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''Homfly polynomial, Homflypt polynomial, skein polynomial''
 
''Homfly polynomial, Homflypt polynomial, skein polynomial''
  
 
An invariant of oriented links.
 
An invariant of oriented links.
  
It is a Laurent polynomial of two variables associated to ambient isotopy classes of oriented links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300401.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300402.png" />), constructed in 1984 by several groups of researchers (thus the acronyms Homfly and Homflypt) [[#References|[a4]]], [[#References|[a3]]], [[#References|[a14]]], [[#References|[a21]]], and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300403.png" />. It generalizes the [[Alexander–Conway polynomial|Alexander–Conway polynomial]] and the Jones polynomial.
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It is a Laurent polynomial of two variables associated to ambient isotopy classes of oriented links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed in 1984 by several groups of researchers (thus the acronyms Homfly and Homflypt) [[#References|[a4]]], [[#References|[a3]]], [[#References|[a14]]], [[#References|[a21]]], and denoted by $P _ { L } ( \square )$. It generalizes the [[Alexander–Conway polynomial|Alexander–Conway polynomial]] and the Jones polynomial.
  
 
There are several constructions of the polynomial, using diagrams of links (and Reidemeister moves, cf. also [[Reidemeister theorem|Reidemeister theorem]]), the braid group (and the Alexander and Markov theorems, cf. also [[Braid theory|Braid theory]]; [[Alexander theorem on braids|Alexander theorem on braids]]; [[Markov braid theorem|Markov braid theorem]]), or statistical mechanics (interpreting the polynomial as a state sum, cf. also [[Statistical mechanics, mathematical problems in|Statistical mechanics, mathematical problems in]]).
 
There are several constructions of the polynomial, using diagrams of links (and Reidemeister moves, cf. also [[Reidemeister theorem|Reidemeister theorem]]), the braid group (and the Alexander and Markov theorems, cf. also [[Braid theory|Braid theory]]; [[Alexander theorem on braids|Alexander theorem on braids]]; [[Markov braid theorem|Markov braid theorem]]), or statistical mechanics (interpreting the polynomial as a state sum, cf. also [[Statistical mechanics, mathematical problems in|Statistical mechanics, mathematical problems in]]).
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The first approach uses the recursive (skein) relation
 
The first approach uses the recursive (skein) relation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300404.png" /></td> </tr></table>
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\begin{equation*} v ^ { - 1 } P _ { L _ { + } } ( v , z ) - v P _ { L_- } ( v , z ) = z P _ { L _ { 0 } } ( v , z ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300407.png" /> form a [[Conway skein triple|Conway skein triple]]. The Jones–Conway polynomial is usually normalized to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300408.png" /> for the trivial knot. Then for the trivial link of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j1300409.png" /> components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004010.png" />, one gets
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where $L _ { + }$, $L_{-}$ and $L_0$ form a [[Conway skein triple|Conway skein triple]]. The Jones–Conway polynomial is usually normalized to be $1$ for the trivial knot. Then for the trivial link of $n$ components, $T _ { n }$, one gets
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004011.png" /></td> </tr></table>
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\begin{equation*} P _ { T _ { n } } ( v , z ) = \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { n - 1 }. \end{equation*}
  
Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004013.png" /> yields the Jones polynomial, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004014.png" />, and substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004016.png" /> yields the [[Alexander–Conway polynomial|Alexander–Conway polynomial]].
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Setting $v = t$ and $z = \sqrt { t } - 1 / \sqrt { t }$ yields the Jones polynomial, $V _ { L } ( t )$, and substituting $v = 1$ and $z = \sqrt { t } - 1 / \sqrt { t }$ yields the [[Alexander–Conway polynomial|Alexander–Conway polynomial]].
  
 
In the second approach one considers the Hecke algebra associated to the Artin braid group and constructs on it the Jones–Ocneanu trace, which essentially is invariant under Markov moves. This approach is strongly related to the first approach, as the Hecke algebra quadratic relation is analogous to the skein relation of the first method.
 
In the second approach one considers the Hecke algebra associated to the Artin braid group and constructs on it the Jones–Ocneanu trace, which essentially is invariant under Markov moves. This approach is strongly related to the first approach, as the Hecke algebra quadratic relation is analogous to the skein relation of the first method.
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004017.png" /> is an element of the Laurent polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004018.png" />. Furthermore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004020.png" /> denotes the number of components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004021.png" />. In particular,
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1) $P _ { L } ( v , z )$ is an element of the Laurent polynomial ring $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$. Furthermore, $( v z ) ^ { \operatorname { com } ( L ) - 1 } P _ { L } ( v , z ) \in \mathbf{Z} [ v ^ { \pm 2 } , z ^ { 2 } ]$, where $\operatorname { com}( L )$ denotes the number of components of $L$. In particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004022.png" /></td> </tr></table>
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\begin{equation*} P _ { L } ( v , z ) = P _ { L } ( - v , - z ) = ( - 1 ) ^ { \operatorname { com } ( L ) - 1 } P _ { L } ( - v , z ). \end{equation*}
  
For example, for the right-handed Hopf link, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004023.png" />, one has
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For example, for the right-handed Hopf link, $2_{1}$, one has
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004024.png" /></td> </tr></table>
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\begin{equation*} P _ { 2 _ { 1 } } = \frac { v - v ^ { 3 } } { z } + v z. \end{equation*}
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004025.png" /> denotes the mirror image of a link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004026.png" />, then
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2) If $\bar{L}$ denotes the mirror image of a link $L$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004027.png" /></td> </tr></table>
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\begin{equation*} P _ { \overline{L} } ( v , z ) = P _ { L } ( - v ^ { - 1 } , z ). \end{equation*}
  
This property often allows one to detect lack of amphicheirality of a link. For example, for the right-handed trefoil knot, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004028.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004029.png" />, so the trefoil knot is not amphicheiral. For the figure eight knot, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004030.png" />, which is amphicheiral, one gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004031.png" />. The first non-amphicheiral knot not detected by the Jones–Conway polynomial is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004032.png" /> knot. The [[Kauffman polynomial|Kauffman polynomial]] also does not detect non-amphicheirality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004033.png" />. However, one can use the Jones–Conway polynomial of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004034.png" />-cable of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004035.png" /> to see that it is non-amphicheiral.
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This property often allows one to detect lack of amphicheirality of a link. For example, for the right-handed trefoil knot, $3_1$, one has $P _ { 3_1 } = 2 v ^ { 2 } - v ^ { 4 } + v ^ { 2 } z ^ { 2 }$, so the trefoil knot is not amphicheiral. For the figure eight knot, $4_1$, which is amphicheiral, one gets $P _ { 4 _ { 1 } } = v ^ { - 2 } - 1 + v ^ { 2 } - z ^ { 2 }$. The first non-amphicheiral knot not detected by the Jones–Conway polynomial is the $9_{42}$ knot. The [[Kauffman polynomial|Kauffman polynomial]] also does not detect non-amphicheirality of $9_{42}$. However, one can use the Jones–Conway polynomial of the $2$-cable of $9_{42}$ to see that it is non-amphicheiral.
  
 
3) Next,
 
3) Next,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004036.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004036.png"/></td> </tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004037.png" /> denotes the [[Connected sum|connected sum]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004038.png" /> the split sum of links. Because the connected sum of links may depend on the choice of connected components, one can use the connected sum formula to find different links with the same Jones–Conway polynomial, for example the connected sum of three Hopf links can give two different results, both with the polynomial equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004039.png" />.
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where $\#$ denotes the [[Connected sum|connected sum]] and $\cup$ the split sum of links. Because the connected sum of links may depend on the choice of connected components, one can use the connected sum formula to find different links with the same Jones–Conway polynomial, for example the connected sum of three Hopf links can give two different results, both with the polynomial equal to $( ( v - v ^ { 3 } ) / z + v z ) ^ { 3 }$.
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004040.png" /> denotes the link obtained by reversing the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004041.png" />, then the Jones–Conway polynomials of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004043.png" /> coincide. Thus, each non-reversible knot (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004044.png" /> is the smallest example) gives rise to an example of different knots with the same Jones–Conway polynomial. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004046.png" /> are two non-reversible knots, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004048.png" /> are different knots which cannot be distinguished by the Jones–Conway polynomials (nor by the Kauffman polynomials) of their satellites. It is an open problem (as of 2001) whether they can be distinguished by any Vassiliev–Gusarov invariants.
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4) If $L^-$ denotes the link obtained by reversing the orientation of $L$, then the Jones–Conway polynomials of $L$ and $L^-$ coincide. Thus, each non-reversible knot ($8 _ { 17 }$ is the smallest example) gives rise to an example of different knots with the same Jones–Conway polynomial. Furthermore, if $K _ { 1 }$ and $K _ { 2 }$ are two non-reversible knots, then $K _ { 1 } \# K _ { 2 }$ and $K _ { 1 } \# K _ { 2 } ^ { - }$ are different knots which cannot be distinguished by the Jones–Conway polynomials (nor by the Kauffman polynomials) of their satellites. It is an open problem (as of 2001) whether they can be distinguished by any Vassiliev–Gusarov invariants.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004049.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004051.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004054.png" /> is any diagram of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004056.png" /> denotes the number of crossings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004058.png" /> is the number of Seifert circles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004059.png" />. The equality holds, e.g., for homogeneous diagrams (including positive and alternating diagrams). In particular for a non-trivial knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004061.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004062.png" /> is the crossing number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004063.png" />, i.e. the minimum over all diagrams of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004064.png" /> of the crossing number.
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If $P _ { L } ( v , z ) = \sum _ { i = m } ^ { M } P _ { i } ( v ) z ^ { i }$ (with $P _ { m } ( v ) \neq 0$, $P _ { M } ( v ) \neq 0$), then $m = 1 - \operatorname { com } ( L )$ and $M \leq \operatorname { cr } ( D _ { L } ) - s ( D _ { L } ) + 1$, where $D_{ L}$ is any diagram of $L$, $\operatorname { cr } ( D _ { L } )$ denotes the number of crossings of $D_{ L}$ and $s ( D _ { L } )$ is the number of Seifert circles of $D_{ L}$. The equality holds, e.g., for homogeneous diagrams (including positive and alternating diagrams). In particular for a non-trivial knot $K$, $M &lt; \text{cr} ( K )$ where $\operatorname { cr } ( K )$ is the crossing number of $K$, i.e. the minimum over all diagrams of $L$ of the crossing number.
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004065.png" />, then
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5) If $\varphi ( D ) = \operatorname { cr } ( D _ { L } ) - s ( D _ { L } ) + 1$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004066.png" /></td> </tr></table>
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\begin{equation*} P _ { \varphi  ( D _ { 1 } * D _ { 2 } )} ( v ) = P _ { \varphi  ( D _ { 1 } )} ( v ) P _ { \varphi  ( D _ { 2 } )} ( v ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004067.png" /> denotes the planar star (Murasugi) product of the diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004069.png" /> [[#References|[a15]]]. D.L. Vertigan proved that for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004071.png" /> can be computed in polynomial time on the number of crossings [[#References|[a20]]].
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where $D _ { 1 } * D _ { 2 }$ denotes the planar star (Murasugi) product of the diagrams $D _ { 1 }$ and $D _ { 2 }$ [[#References|[a15]]]. D.L. Vertigan proved that for a fixed $i$, $P _ { i } ( v )$ can be computed in polynomial time on the number of crossings [[#References|[a20]]].
  
6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004072.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004074.png" />), then
+
6) If $P _ { L } ( v , z ) = \sum _ { i = e } ^ { E } a _ { i } ( z ) v ^ { i }$ (with $\alpha _ { e} ( z ) \neq 0$, $a _ { E } ( z ) \neq 0$), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004075.png" /></td> </tr></table>
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\begin{equation*} ( n _ { + } - n _ { - } ) - ( s ( D _ { L } ) - 1 ) \leq e \leq E \leq ( n _ { + } - n _ { - } ) + ( s ( D _ { L } ) - 1 ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004076.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004077.png" />) is the number of positive (respectively, negative) crossings of the diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004078.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004079.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004080.png" /> is the minimal number of Seifert circles of all diagrams <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004081.png" /> representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004082.png" />; it is equal to the braid index of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004083.png" />, see [[#References|[a24]]]. The knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004084.png" /> is the first knot for which the inequality is sharp. The smallest (known) alternating knot for which the inequality is sharp has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004085.png" /> crossings [[#References|[a16]]].
+
where $n _ { + }$ (respectively, $n_-$) is the number of positive (respectively, negative) crossings of the diagram $D_{ L}$. In particular, $s ( L ) \geq ( E - e ) / 2$, where $s ( L )$ is the minimal number of Seifert circles of all diagrams $D_{ L}$ representing $L$; it is equal to the braid index of $L$, see [[#References|[a24]]]. The knot $9_{42}$ is the first knot for which the inequality is sharp. The smallest (known) alternating knot for which the inequality is sharp has $18$ crossings [[#References|[a16]]].
  
7) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004086.png" />, then
+
7) If $P _ { L } ( v , z ) = \sum a _ { i ,j} v ^ { i } z ^ { j }$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004087.png" /></td> </tr></table>
+
\begin{equation*} a_{ ( n _ { + } - n _ { - } - s ( D _ { L } ) + 1 ) , ( n - s ( D _ { L } ) + 1 )  } \neq 0 \end{equation*}
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004088.png" /> is a positive diagram (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004089.png" />). For example, for a positive diagram of the right-handed trefoil knot with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004090.png" /> crossings one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004091.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004092.png" />.
+
if and only if $D_{ L}$ is a positive diagram (i.e. $n_{- } = 0$). For example, for a positive diagram of the right-handed trefoil knot with $3$ crossings one has $s ( D _ { 3_{1} } ) = 2$ and $a _ { 2  , 2} = 1$.
  
8) For a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004094.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004095.png" />.
+
8) For a knot $K$, $P _ { K } ( v , z ) - 1$ is a multiple of $( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 }$.
  
For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004096.png" />. Setting
+
For example, $P _ { 4 _ { 1 } } ( v , z ) - 1 = ( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 } = - v ^ { - 2 } ( P _ { 3_1 } ( v , z ) - 1 ) = - v ^ { 2 } ( P _ { \overline{3}_1 } ( v , z ) - 1 )$. Setting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004097.png" /></td> </tr></table>
+
\begin{equation*} \mathsf{P} _ { K } ( v , z ) = \frac { P _ { K } ( v , z ) - 1 } { ( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 } }, \end{equation*}
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004098.png" /></td> </tr></table>
+
\begin{equation*} \mathsf{P} _ { K _ { + } } ( v , z ) - \mathsf{P} _ { K _ { - } } ( v , z ) \equiv \operatorname { lk } ( K _ { 0 } ) \operatorname { mod } ( v ^ { 2 } - 1 , z ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j13004099.png" /> is the linking number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040100.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040101.png" /> (the second coefficient of the [[Alexander–Conway polynomial|Alexander–Conway polynomial]] and the first non-trivial Vassiliev–Gusarov invariant). More generally, consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040102.png" /> as an element of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040103.png" /> that is the subring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040104.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040107.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040108.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040109.png" />. Furthermore,
+
where $\operatorname { lk } ( K _ { 0 } )$ is the linking number of $K _ { 0 }$. In particular, $\mathsf{P} _ { K } ( 1,0 ) = a _ { 2 }$ (the second coefficient of the [[Alexander–Conway polynomial|Alexander–Conway polynomial]] and the first non-trivial Vassiliev–Gusarov invariant). More generally, consider $P _ { L } ( v , z )$ as an element of the ring $R$ that is the subring of $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$ generated by $v \pm 1$, $z$ and $( v ^ { - 1 } - v ) / z$. Then $P _ { L } ( v , z ) - P _ { T  _ { \text{com} ( L ) }} ( v , z )$ is a multiple of $z ( ( ( v ^ { - 1 } - v ) / z ) ^ { 2 } - 1 )$. Furthermore,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040110.png" /></td> </tr></table>
+
\begin{equation*} \frac { P _ { L } ( v , z ) - P _ { T_{\operatorname{ com } ( L )}}  ( v , z ) } { z \left( \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { 2 } - 1 \right) } \equiv \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040111.png" /></td> </tr></table>
+
\begin{equation*} \equiv - \operatorname { lk } ( L ) v \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { \operatorname { com } ( L ) - 2 } \operatorname { mod } ( z ) \end{equation*}
  
in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040112.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040113.png" /> is the linking number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040114.png" />. In particular,
+
in $R$, where $\operatorname { lk } ( L )$ is the linking number of $L$. In particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040115.png" /></td> </tr></table>
+
\begin{equation*} \frac { P _ { 2_1 } ( v , z ) - \frac { v ^ { - 1 } - v } { z } } { z \left( \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { 2 } - 1 \right) } = - v. \end{equation*}
  
 
The number of components of a link and its linking number can be recovered from the Jones–Conway polynomial.
 
The number of components of a link and its linking number can be recovered from the Jones–Conway polynomial.
  
9) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040116.png" /> for a knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040117.png" /> and some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040118.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040119.png" /> is a Vassiliev–Gusarov invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040120.png" />. Equivalently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040121.png" /> is a Vassiliev–Gusarov invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040122.png" />. One can obtain a coloured Jones–Conway polynomial by choosing an element of the Jones–Conway [[Skein module|skein module]] of the solid torus (a linear combination of links in a solid torus) and computing for this the Jones–Conway polynomial. Such an invariant is stratified by Vassiliev–Gusarov invariants.
+
9) If $P _ { K } ( v , z ) = v ^ { 2 c } \sum  c _ { i, j } ( v ^ { 2 } - 1 ) ^ { i } z ^ { j }$ for a knot $K$ and some constant $c$, then $c_{i , j}$ is a Vassiliev–Gusarov invariant of order $i + j$. Equivalently, $P _ { K } ( v , z ) \operatorname { mod } ( ( ( v ^ { 2 } - 1 ) , z ) ^ { k + 1 } )$ is a Vassiliev–Gusarov invariant of order $k$. One can obtain a coloured Jones–Conway polynomial by choosing an element of the Jones–Conway [[Skein module|skein module]] of the solid torus (a linear combination of links in a solid torus) and computing for this the Jones–Conway polynomial. Such an invariant is stratified by Vassiliev–Gusarov invariants.
  
Computing the whole Jones–Conway polynomial is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040124.png" />-hard [[#References|[a6]]] (so up to the conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040125.png" />, the polynomial cannot be computed in polynomial time). Furthermore, computing most of the substitutions to the Jones–Conway polynomial is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040126.png" />-hard (even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040127.png" />-hard). The exceptions have well understood interpretations [[#References|[a5]]]:
+
Computing the whole Jones–Conway polynomial is $\cal N P$-hard [[#References|[a6]]] (so up to the conjecture that $\mathcal{N P} \neq \mathcal{P}$, the polynomial cannot be computed in polynomial time). Furthermore, computing most of the substitutions to the Jones–Conway polynomial is $\cal N P$-hard (even $\# \mathcal{P}$-hard). The exceptions have well understood interpretations [[#References|[a5]]]:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040128.png" />. Now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040129.png" /> reduces to the [[Alexander–Conway polynomial|Alexander–Conway polynomial]].
+
i) $v = \pm 1$. Now $P _ { L }$ reduces to the [[Alexander–Conway polynomial|Alexander–Conway polynomial]].
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040130.png" />. Compare 8) above.
+
ii) $z = \pm ( v ^ { - 1 } - v )$. Compare 8) above.
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040131.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040132.png" />'s are independent. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040133.png" /> if the [[Arf-invariant|Arf-invariant]] is defined and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040134.png" /> otherwise.
+
iii) $( v , z ) = ( \pm i , \pm i \sqrt { 2 } )$, where the $\pm$'s are independent. For example, $P ( i , i \sqrt { 2 } ) = ( - \sqrt { 2 } ) ^ { \operatorname { com } ( L ) - 1 } ( - 1 ) ^ { \operatorname { Arf } ( L ) }$ if the [[Arf-invariant|Arf-invariant]] is defined and $0$ otherwise.
  
iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040135.png" />. E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040136.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040137.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040138.png" />-fold cyclic covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040139.png" /> branched over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040140.png" />.
+
iv) $( v , z ) = ( \pm i , \pm i )$. E.g., $P _ { L } ( i , i ) = ( i \sqrt { 2 } ) ^ { \operatorname { dim } ( H _ { 1 } ( M ^ { ( 3 ) } , \mathbf{Z} _ { 2 } ) ) }$, where $M ^ { ( k ) }$ denotes the $k$-fold cyclic covering of $S ^ { 3 }$ branched over $L$.
  
v) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040141.png" />. E.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040143.png" /> can be derived from the modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040144.png" /> linking form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040145.png" />.
+
v) $( v , z ) = ( \pm e ^ { \pm \pi i / 3 } , \pm i )$. E.g., $P _ { L } ( e ^ { \pi i / 3 } , i ) = \varepsilon ( L ) i ^ { \operatorname { com } ( L ) - 1 } ( i \sqrt { 3 } ) ^ { \operatorname { dim } ( H _ { 1 } ( M ^ { ( 2 ) } , \mathbf{Z} _ { 3 } ) ) }$, where $\varepsilon ( L ) = \pm 1$ can be derived from the modulo $3$ linking form in $M ^ { ( 2 ) }$.
  
 
There are several constructions of different links with the same Jones–Conway polynomial: mutation, rotation, cabling, spectral parameter tangle, etc. [[#References|[a22]]]. T. Kanenobu [[#References|[a12]]] has constructed an infinite family of different links with the same Jones–Conway polynomial. It is an open problem (as of 2001) whether there exists a non-trivial link with Jones–Conway polynomial equal to that of a trivial link (cf. [[Jones unknotting conjecture|Jones unknotting conjecture]]).
 
There are several constructions of different links with the same Jones–Conway polynomial: mutation, rotation, cabling, spectral parameter tangle, etc. [[#References|[a22]]]. T. Kanenobu [[#References|[a12]]] has constructed an infinite family of different links with the same Jones–Conway polynomial. It is an open problem (as of 2001) whether there exists a non-trivial link with Jones–Conway polynomial equal to that of a trivial link (cf. [[Jones unknotting conjecture|Jones unknotting conjecture]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Cromwell,  "Homogeneous links"  ''J. London Math. Soc.'' , '''39''' :  2  (1989)  pp. 535–552</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Franks,  R.F. Williams,  "Braids and the Jones polynomial"  ''Trans. Amer. Math. Soc.'' , '''303'''  (1987)  pp. 97–108</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Hoste,  "A polynomial invariant of knots and links"  ''Pacific J. Math.'' , '''124'''  (1986)  pp. 295–320</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Freyd,  D. Yetter,  J. Hoste,  W.B.R. Lickorish,  K. Millett,  A. Ocneanu,  "A new polynomial invariant of knots and links"  ''Bull. Amer. Math. Soc.'' , '''12'''  (1985)  pp. 239–249</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Jaeger,  D.L. Vertigan,  D.J.A. Welsh,  "On the computational complexity of the Jones and Tutte polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 35–53</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Jaeger,  "On Tutte polynomials and link polynomials"  ''Proc. Amer. Math. Soc.'' , '''103''' :  2  (1988)  pp. 647–654</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F. Jaeger,  "Composition products and models for the Homfly polynomial"  ''L'Enseign. Math.'' , '''35'''  (1989)  pp. 323–361</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  V.F.R. Jones,  "Hecke algebra representations of braid groups and link polynomials"  ''Ann. of Math.'' , '''126''' :  2  (1987)  pp. 335–388</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V.F.R. Jones,  "On knot invariants related to some statistical mechanical models"  ''Pacific J. Math.'' , '''137''' :  2  (1989)  pp. 311–334</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  K. Kobayashi,  K. Kodama,  "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040146.png" /> for plumbing diagrams and oriented arborescent links"  ''Kobe J. Math.'' , '''5'''  (1988)  pp. 221–232</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J. Kania–Bartoszyńska,  J.H. Przytycki,  "Knots and links, revisited"  ''Delta, Warsaw'' , '''June'''  (1985)  pp. 10–12  (In Polish)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  T. Kanenobu,  "Infinitely many knots with the same polynomial invariant"  ''Proc. Amer. Math. Soc.'' , '''97''' :  1  (1986)  pp. 158–162</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W.B.R. Lickorish,  "An introduction to knot theory" , Springer  (1997)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  W.B.R. Lickorish,  K. Millett,  "A polynomial invariant of oriented links"  ''Topology'' , '''26'''  (1987)  pp. 107–141</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  K. Murasugi,  J.H. Przytycki,  "The Skein polynomial of a planar star product of two links"  ''Math. Proc. Cambridge Philos. Soc.'' , '''106'''  (1989)  pp. 273–276</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  K. Murasugi,  J.H. Przytycki,  "An index of a graph with applications to knot theory" , ''Memoirs'' , '''106''' , Amer. Math. Soc.  (1993)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  H.R. Morton,  H.B. Short,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j130/j130040/j130040147.png" />-variable polynomial of cable knots"  ''Math. Proc. Cambridge Philos. Soc.'' , '''101'''  (1987)  pp. 267–278</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  H.R. Morton,  "Seifert circles and knot polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''99'''  (1986)  pp. 107–109</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  H. Murakami,  "On derivatives of the Jones polynomial"  ''Kobe J. Math.'' , '''3'''  (1986)  pp. 61–64</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  T.M. Przytycka,  J.H. Przytycki,  "Subexponentially computable truncations of Jones-type polynomials" , ''Graph Structure Theory'' , ''Contemp. Math.'' , '''147'''  (1993)  pp. 63–108</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  J.H. Przytycki,  P. Traczyk,  "Invariants of links of Conway type"  ''Kobe J. Math.'' , '''4'''  (1987)  pp. 115–139</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  J.H. Przytycki,  "Search for different links with the same Jones' type polynomials" , ''Ideas from Graph Theory and Statistical Mechanics, Panoramas of Mathematics'' , '''34''' , Banach Center Publ.  (1995)  pp. 121–148</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  V.G. Turaev,  "The Yang–Baxter equation and invariants of links"  ''Invent. Math.'' , '''92'''  (1988)  pp. 527–553</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  S. Yamada,  "The minimal number of Seifert circles equals to braid index of a link"  ''Invent. Math.'' , '''89'''  (1987)  pp. 347–356</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  P.R. Cromwell,  "Homogeneous links"  ''J. London Math. Soc.'' , '''39''' :  2  (1989)  pp. 535–552</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Franks,  R.F. Williams,  "Braids and the Jones polynomial"  ''Trans. Amer. Math. Soc.'' , '''303'''  (1987)  pp. 97–108</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Hoste,  "A polynomial invariant of knots and links"  ''Pacific J. Math.'' , '''124'''  (1986)  pp. 295–320</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Freyd,  D. Yetter,  J. Hoste,  W.B.R. Lickorish,  K. Millett,  A. Ocneanu,  "A new polynomial invariant of knots and links"  ''Bull. Amer. Math. Soc.'' , '''12'''  (1985)  pp. 239–249</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  F. Jaeger,  D.L. Vertigan,  D.J.A. Welsh,  "On the computational complexity of the Jones and Tutte polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 35–53</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  F. Jaeger,  "On Tutte polynomials and link polynomials"  ''Proc. Amer. Math. Soc.'' , '''103''' :  2  (1988)  pp. 647–654</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  F. Jaeger,  "Composition products and models for the Homfly polynomial"  ''L'Enseign. Math.'' , '''35'''  (1989)  pp. 323–361</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  V.F.R. Jones,  "Hecke algebra representations of braid groups and link polynomials"  ''Ann. of Math.'' , '''126''' :  2  (1987)  pp. 335–388</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  V.F.R. Jones,  "On knot invariants related to some statistical mechanical models"  ''Pacific J. Math.'' , '''137''' :  2  (1989)  pp. 311–334</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  K. Kobayashi,  K. Kodama,  "On the $\operatorname { deg } _ { z } P _ { L } ( v , z )$ for plumbing diagrams and oriented arborescent links"  ''Kobe J. Math.'' , '''5'''  (1988)  pp. 221–232</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J. Kania–Bartoszyńska,  J.H. Przytycki,  "Knots and links, revisited"  ''Delta, Warsaw'' , '''June'''  (1985)  pp. 10–12  (In Polish)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  T. Kanenobu,  "Infinitely many knots with the same polynomial invariant"  ''Proc. Amer. Math. Soc.'' , '''97''' :  1  (1986)  pp. 158–162</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  W.B.R. Lickorish,  "An introduction to knot theory" , Springer  (1997)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  W.B.R. Lickorish,  K. Millett,  "A polynomial invariant of oriented links"  ''Topology'' , '''26'''  (1987)  pp. 107–141</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  K. Murasugi,  J.H. Przytycki,  "The Skein polynomial of a planar star product of two links"  ''Math. Proc. Cambridge Philos. Soc.'' , '''106'''  (1989)  pp. 273–276</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  K. Murasugi,  J.H. Przytycki,  "An index of a graph with applications to knot theory" , ''Memoirs'' , '''106''' , Amer. Math. Soc.  (1993)</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  H.R. Morton,  H.B. Short,  "The $2$-variable polynomial of cable knots"  ''Math. Proc. Cambridge Philos. Soc.'' , '''101'''  (1987)  pp. 267–278</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  H.R. Morton,  "Seifert circles and knot polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''99'''  (1986)  pp. 107–109</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  H. Murakami,  "On derivatives of the Jones polynomial"  ''Kobe J. Math.'' , '''3'''  (1986)  pp. 61–64</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  T.M. Przytycka,  J.H. Przytycki,  "Subexponentially computable truncations of Jones-type polynomials" , ''Graph Structure Theory'' , ''Contemp. Math.'' , '''147'''  (1993)  pp. 63–108</td></tr><tr><td valign="top">[a21]</td> <td valign="top">  J.H. Przytycki,  P. Traczyk,  "Invariants of links of Conway type"  ''Kobe J. Math.'' , '''4'''  (1987)  pp. 115–139</td></tr><tr><td valign="top">[a22]</td> <td valign="top">  J.H. Przytycki,  "Search for different links with the same Jones' type polynomials" , ''Ideas from Graph Theory and Statistical Mechanics, Panoramas of Mathematics'' , '''34''' , Banach Center Publ.  (1995)  pp. 121–148</td></tr><tr><td valign="top">[a23]</td> <td valign="top">  V.G. Turaev,  "The Yang–Baxter equation and invariants of links"  ''Invent. Math.'' , '''92'''  (1988)  pp. 527–553</td></tr><tr><td valign="top">[a24]</td> <td valign="top">  S. Yamada,  "The minimal number of Seifert circles equals to braid index of a link"  ''Invent. Math.'' , '''89'''  (1987)  pp. 347–356</td></tr></table>

Revision as of 16:58, 1 July 2020

Homfly polynomial, Homflypt polynomial, skein polynomial

An invariant of oriented links.

It is a Laurent polynomial of two variables associated to ambient isotopy classes of oriented links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed in 1984 by several groups of researchers (thus the acronyms Homfly and Homflypt) [a4], [a3], [a14], [a21], and denoted by $P _ { L } ( \square )$. It generalizes the Alexander–Conway polynomial and the Jones polynomial.

There are several constructions of the polynomial, using diagrams of links (and Reidemeister moves, cf. also Reidemeister theorem), the braid group (and the Alexander and Markov theorems, cf. also Braid theory; Alexander theorem on braids; Markov braid theorem), or statistical mechanics (interpreting the polynomial as a state sum, cf. also Statistical mechanics, mathematical problems in).

The first approach uses the recursive (skein) relation

\begin{equation*} v ^ { - 1 } P _ { L _ { + } } ( v , z ) - v P _ { L_- } ( v , z ) = z P _ { L _ { 0 } } ( v , z ), \end{equation*}

where $L _ { + }$, $L_{-}$ and $L_0$ form a Conway skein triple. The Jones–Conway polynomial is usually normalized to be $1$ for the trivial knot. Then for the trivial link of $n$ components, $T _ { n }$, one gets

\begin{equation*} P _ { T _ { n } } ( v , z ) = \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { n - 1 }. \end{equation*}

Setting $v = t$ and $z = \sqrt { t } - 1 / \sqrt { t }$ yields the Jones polynomial, $V _ { L } ( t )$, and substituting $v = 1$ and $z = \sqrt { t } - 1 / \sqrt { t }$ yields the Alexander–Conway polynomial.

In the second approach one considers the Hecke algebra associated to the Artin braid group and constructs on it the Jones–Ocneanu trace, which essentially is invariant under Markov moves. This approach is strongly related to the first approach, as the Hecke algebra quadratic relation is analogous to the skein relation of the first method.

The third approach uses the fact that the statistical mechanical systems considered satisfy the Yang–Baxter equation. This approach is immediately related to the first two by the fact that a Yang–Baxter operator satisfies the minimal polynomial (leading to a skein relation) and, on the other hand, yields a linear representation of the braid group. This state sum approach was first developed by V.F.R. Jones and has a very nice reflection in the Jaeger composition product [a7].

Properties of the Jones–Conway polynomial.

1) $P _ { L } ( v , z )$ is an element of the Laurent polynomial ring $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$. Furthermore, $( v z ) ^ { \operatorname { com } ( L ) - 1 } P _ { L } ( v , z ) \in \mathbf{Z} [ v ^ { \pm 2 } , z ^ { 2 } ]$, where $\operatorname { com}( L )$ denotes the number of components of $L$. In particular,

\begin{equation*} P _ { L } ( v , z ) = P _ { L } ( - v , - z ) = ( - 1 ) ^ { \operatorname { com } ( L ) - 1 } P _ { L } ( - v , z ). \end{equation*}

For example, for the right-handed Hopf link, $2_{1}$, one has

\begin{equation*} P _ { 2 _ { 1 } } = \frac { v - v ^ { 3 } } { z } + v z. \end{equation*}

2) If $\bar{L}$ denotes the mirror image of a link $L$, then

\begin{equation*} P _ { \overline{L} } ( v , z ) = P _ { L } ( - v ^ { - 1 } , z ). \end{equation*}

This property often allows one to detect lack of amphicheirality of a link. For example, for the right-handed trefoil knot, $3_1$, one has $P _ { 3_1 } = 2 v ^ { 2 } - v ^ { 4 } + v ^ { 2 } z ^ { 2 }$, so the trefoil knot is not amphicheiral. For the figure eight knot, $4_1$, which is amphicheiral, one gets $P _ { 4 _ { 1 } } = v ^ { - 2 } - 1 + v ^ { 2 } - z ^ { 2 }$. The first non-amphicheiral knot not detected by the Jones–Conway polynomial is the $9_{42}$ knot. The Kauffman polynomial also does not detect non-amphicheirality of $9_{42}$. However, one can use the Jones–Conway polynomial of the $2$-cable of $9_{42}$ to see that it is non-amphicheiral.

3) Next,

where $\#$ denotes the connected sum and $\cup$ the split sum of links. Because the connected sum of links may depend on the choice of connected components, one can use the connected sum formula to find different links with the same Jones–Conway polynomial, for example the connected sum of three Hopf links can give two different results, both with the polynomial equal to $( ( v - v ^ { 3 } ) / z + v z ) ^ { 3 }$.

4) If $L^-$ denotes the link obtained by reversing the orientation of $L$, then the Jones–Conway polynomials of $L$ and $L^-$ coincide. Thus, each non-reversible knot ($8 _ { 17 }$ is the smallest example) gives rise to an example of different knots with the same Jones–Conway polynomial. Furthermore, if $K _ { 1 }$ and $K _ { 2 }$ are two non-reversible knots, then $K _ { 1 } \# K _ { 2 }$ and $K _ { 1 } \# K _ { 2 } ^ { - }$ are different knots which cannot be distinguished by the Jones–Conway polynomials (nor by the Kauffman polynomials) of their satellites. It is an open problem (as of 2001) whether they can be distinguished by any Vassiliev–Gusarov invariants.

If $P _ { L } ( v , z ) = \sum _ { i = m } ^ { M } P _ { i } ( v ) z ^ { i }$ (with $P _ { m } ( v ) \neq 0$, $P _ { M } ( v ) \neq 0$), then $m = 1 - \operatorname { com } ( L )$ and $M \leq \operatorname { cr } ( D _ { L } ) - s ( D _ { L } ) + 1$, where $D_{ L}$ is any diagram of $L$, $\operatorname { cr } ( D _ { L } )$ denotes the number of crossings of $D_{ L}$ and $s ( D _ { L } )$ is the number of Seifert circles of $D_{ L}$. The equality holds, e.g., for homogeneous diagrams (including positive and alternating diagrams). In particular for a non-trivial knot $K$, $M < \text{cr} ( K )$ where $\operatorname { cr } ( K )$ is the crossing number of $K$, i.e. the minimum over all diagrams of $L$ of the crossing number.

5) If $\varphi ( D ) = \operatorname { cr } ( D _ { L } ) - s ( D _ { L } ) + 1$, then

\begin{equation*} P _ { \varphi ( D _ { 1 } * D _ { 2 } )} ( v ) = P _ { \varphi ( D _ { 1 } )} ( v ) P _ { \varphi ( D _ { 2 } )} ( v ), \end{equation*}

where $D _ { 1 } * D _ { 2 }$ denotes the planar star (Murasugi) product of the diagrams $D _ { 1 }$ and $D _ { 2 }$ [a15]. D.L. Vertigan proved that for a fixed $i$, $P _ { i } ( v )$ can be computed in polynomial time on the number of crossings [a20].

6) If $P _ { L } ( v , z ) = \sum _ { i = e } ^ { E } a _ { i } ( z ) v ^ { i }$ (with $\alpha _ { e} ( z ) \neq 0$, $a _ { E } ( z ) \neq 0$), then

\begin{equation*} ( n _ { + } - n _ { - } ) - ( s ( D _ { L } ) - 1 ) \leq e \leq E \leq ( n _ { + } - n _ { - } ) + ( s ( D _ { L } ) - 1 ), \end{equation*}

where $n _ { + }$ (respectively, $n_-$) is the number of positive (respectively, negative) crossings of the diagram $D_{ L}$. In particular, $s ( L ) \geq ( E - e ) / 2$, where $s ( L )$ is the minimal number of Seifert circles of all diagrams $D_{ L}$ representing $L$; it is equal to the braid index of $L$, see [a24]. The knot $9_{42}$ is the first knot for which the inequality is sharp. The smallest (known) alternating knot for which the inequality is sharp has $18$ crossings [a16].

7) If $P _ { L } ( v , z ) = \sum a _ { i ,j} v ^ { i } z ^ { j }$, then

\begin{equation*} a_{ ( n _ { + } - n _ { - } - s ( D _ { L } ) + 1 ) , ( n - s ( D _ { L } ) + 1 ) } \neq 0 \end{equation*}

if and only if $D_{ L}$ is a positive diagram (i.e. $n_{- } = 0$). For example, for a positive diagram of the right-handed trefoil knot with $3$ crossings one has $s ( D _ { 3_{1} } ) = 2$ and $a _ { 2 , 2} = 1$.

8) For a knot $K$, $P _ { K } ( v , z ) - 1$ is a multiple of $( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 }$.

For example, $P _ { 4 _ { 1 } } ( v , z ) - 1 = ( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 } = - v ^ { - 2 } ( P _ { 3_1 } ( v , z ) - 1 ) = - v ^ { 2 } ( P _ { \overline{3}_1 } ( v , z ) - 1 )$. Setting

\begin{equation*} \mathsf{P} _ { K } ( v , z ) = \frac { P _ { K } ( v , z ) - 1 } { ( v ^ { - 1 } - v ) ^ { 2 } - z ^ { 2 } }, \end{equation*}

then

\begin{equation*} \mathsf{P} _ { K _ { + } } ( v , z ) - \mathsf{P} _ { K _ { - } } ( v , z ) \equiv \operatorname { lk } ( K _ { 0 } ) \operatorname { mod } ( v ^ { 2 } - 1 , z ), \end{equation*}

where $\operatorname { lk } ( K _ { 0 } )$ is the linking number of $K _ { 0 }$. In particular, $\mathsf{P} _ { K } ( 1,0 ) = a _ { 2 }$ (the second coefficient of the Alexander–Conway polynomial and the first non-trivial Vassiliev–Gusarov invariant). More generally, consider $P _ { L } ( v , z )$ as an element of the ring $R$ that is the subring of $\mathbf{Z} [ v ^ { \pm 1 } , z ^ { \pm 1 } ]$ generated by $v \pm 1$, $z$ and $( v ^ { - 1 } - v ) / z$. Then $P _ { L } ( v , z ) - P _ { T _ { \text{com} ( L ) }} ( v , z )$ is a multiple of $z ( ( ( v ^ { - 1 } - v ) / z ) ^ { 2 } - 1 )$. Furthermore,

\begin{equation*} \frac { P _ { L } ( v , z ) - P _ { T_{\operatorname{ com } ( L )}} ( v , z ) } { z \left( \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { 2 } - 1 \right) } \equiv \end{equation*}

\begin{equation*} \equiv - \operatorname { lk } ( L ) v \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { \operatorname { com } ( L ) - 2 } \operatorname { mod } ( z ) \end{equation*}

in $R$, where $\operatorname { lk } ( L )$ is the linking number of $L$. In particular,

\begin{equation*} \frac { P _ { 2_1 } ( v , z ) - \frac { v ^ { - 1 } - v } { z } } { z \left( \left( \frac { v ^ { - 1 } - v } { z } \right) ^ { 2 } - 1 \right) } = - v. \end{equation*}

The number of components of a link and its linking number can be recovered from the Jones–Conway polynomial.

9) If $P _ { K } ( v , z ) = v ^ { 2 c } \sum c _ { i, j } ( v ^ { 2 } - 1 ) ^ { i } z ^ { j }$ for a knot $K$ and some constant $c$, then $c_{i , j}$ is a Vassiliev–Gusarov invariant of order $i + j$. Equivalently, $P _ { K } ( v , z ) \operatorname { mod } ( ( ( v ^ { 2 } - 1 ) , z ) ^ { k + 1 } )$ is a Vassiliev–Gusarov invariant of order $k$. One can obtain a coloured Jones–Conway polynomial by choosing an element of the Jones–Conway skein module of the solid torus (a linear combination of links in a solid torus) and computing for this the Jones–Conway polynomial. Such an invariant is stratified by Vassiliev–Gusarov invariants.

Computing the whole Jones–Conway polynomial is $\cal N P$-hard [a6] (so up to the conjecture that $\mathcal{N P} \neq \mathcal{P}$, the polynomial cannot be computed in polynomial time). Furthermore, computing most of the substitutions to the Jones–Conway polynomial is $\cal N P$-hard (even $\# \mathcal{P}$-hard). The exceptions have well understood interpretations [a5]:

i) $v = \pm 1$. Now $P _ { L }$ reduces to the Alexander–Conway polynomial.

ii) $z = \pm ( v ^ { - 1 } - v )$. Compare 8) above.

iii) $( v , z ) = ( \pm i , \pm i \sqrt { 2 } )$, where the $\pm$'s are independent. For example, $P ( i , i \sqrt { 2 } ) = ( - \sqrt { 2 } ) ^ { \operatorname { com } ( L ) - 1 } ( - 1 ) ^ { \operatorname { Arf } ( L ) }$ if the Arf-invariant is defined and $0$ otherwise.

iv) $( v , z ) = ( \pm i , \pm i )$. E.g., $P _ { L } ( i , i ) = ( i \sqrt { 2 } ) ^ { \operatorname { dim } ( H _ { 1 } ( M ^ { ( 3 ) } , \mathbf{Z} _ { 2 } ) ) }$, where $M ^ { ( k ) }$ denotes the $k$-fold cyclic covering of $S ^ { 3 }$ branched over $L$.

v) $( v , z ) = ( \pm e ^ { \pm \pi i / 3 } , \pm i )$. E.g., $P _ { L } ( e ^ { \pi i / 3 } , i ) = \varepsilon ( L ) i ^ { \operatorname { com } ( L ) - 1 } ( i \sqrt { 3 } ) ^ { \operatorname { dim } ( H _ { 1 } ( M ^ { ( 2 ) } , \mathbf{Z} _ { 3 } ) ) }$, where $\varepsilon ( L ) = \pm 1$ can be derived from the modulo $3$ linking form in $M ^ { ( 2 ) }$.

There are several constructions of different links with the same Jones–Conway polynomial: mutation, rotation, cabling, spectral parameter tangle, etc. [a22]. T. Kanenobu [a12] has constructed an infinite family of different links with the same Jones–Conway polynomial. It is an open problem (as of 2001) whether there exists a non-trivial link with Jones–Conway polynomial equal to that of a trivial link (cf. Jones unknotting conjecture).

References

[a1] P.R. Cromwell, "Homogeneous links" J. London Math. Soc. , 39 : 2 (1989) pp. 535–552
[a2] J. Franks, R.F. Williams, "Braids and the Jones polynomial" Trans. Amer. Math. Soc. , 303 (1987) pp. 97–108
[a3] J. Hoste, "A polynomial invariant of knots and links" Pacific J. Math. , 124 (1986) pp. 295–320
[a4] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, "A new polynomial invariant of knots and links" Bull. Amer. Math. Soc. , 12 (1985) pp. 239–249
[a5] F. Jaeger, D.L. Vertigan, D.J.A. Welsh, "On the computational complexity of the Jones and Tutte polynomials" Math. Proc. Cambridge Philos. Soc. , 108 (1990) pp. 35–53
[a6] F. Jaeger, "On Tutte polynomials and link polynomials" Proc. Amer. Math. Soc. , 103 : 2 (1988) pp. 647–654
[a7] F. Jaeger, "Composition products and models for the Homfly polynomial" L'Enseign. Math. , 35 (1989) pp. 323–361
[a8] V.F.R. Jones, "Hecke algebra representations of braid groups and link polynomials" Ann. of Math. , 126 : 2 (1987) pp. 335–388
[a9] V.F.R. Jones, "On knot invariants related to some statistical mechanical models" Pacific J. Math. , 137 : 2 (1989) pp. 311–334
[a10] K. Kobayashi, K. Kodama, "On the $\operatorname { deg } _ { z } P _ { L } ( v , z )$ for plumbing diagrams and oriented arborescent links" Kobe J. Math. , 5 (1988) pp. 221–232
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How to Cite This Entry:
Jones-Conway polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jones-Conway_polynomial&oldid=50281
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article