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An invariant of unoriented framed links.
 
An invariant of unoriented framed links.
  
It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300101.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300102.png" />), constructed by L.H. Kauffman in the summer of 1985 and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300103.png" />. It is defined recursively as follows: For a trivial link of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300104.png" /> components, with zero framing, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300105.png" />, one puts
+
It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed by L.H. Kauffman in the summer of 1985 and denoted by $\langle L \rangle$. It is defined recursively as follows: For a trivial link of $n$ components, with zero framing, $T _ { n }$, one puts
  
<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/k/k130/k130010/k1300106.png" /></td> </tr></table>
+
\begin{equation*} \langle T _ { n } \rangle = ( - A ^ { 2 } - A ^ { - 2 } ) ^ { n - 1 }. \end{equation*}
  
 
For the Kauffman bracket skein triple (cf. Fig.a1) one has the Kauffman bracket skein relation:
 
For the Kauffman bracket skein triple (cf. Fig.a1) one has the Kauffman bracket 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/k/k130/k130010/k1300107.png" /></td> </tr></table>
+
\begin{equation*} \langle L _ { + } \rangle = A \langle L _ { 0 } \rangle + A ^ { - 1 } \langle L _ { \infty } \rangle . \end{equation*}
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k130010a.gif" />
+
<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k130010a.gif" style="border:1px solid;"/>
  
 
Figure: k130010a
 
Figure: k130010a
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300108.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k1300109.png" /> by a positive full twist on its framing, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001010.png" />. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001012.png" />, cf. [[Reidemeister theorem|Reidemeister theorem]]) of diagrams on the plane. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001013.png" /> is changed by the first Reidemeister move by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001014.png" />.
+
If $L ^ { ( 1 ) }$ is obtained from $L$ by a positive full twist on its framing, then $\langle L ^ { ( 1 ) } \rangle = - A ^ { 3 } \langle L \rangle$. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves: $R _ { 2 }$, $R _ { 3 }$, cf. [[Reidemeister theorem|Reidemeister theorem]]) of diagrams on the plane. $\langle D \rangle$ is changed by the first Reidemeister move by $- A ^ { \pm 3 }$.
  
 
The Kauffman bracket polynomial is related to a substitution of the dichromatic polynomial of signed graphs. This connection also relates the Kauffman bracket polynomial to the Potts model in statistical mechanics. State sum expansions of the dichromatic polynomial have their analogue for the Kauffman bracket polynomial. For example:
 
The Kauffman bracket polynomial is related to a substitution of the dichromatic polynomial of signed graphs. This connection also relates the Kauffman bracket polynomial to the Potts model in statistical mechanics. State sum expansions of the dichromatic polynomial have their analogue for the Kauffman bracket polynomial. For example:
  
<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/k/k130/k130010/k13001015.png" /></td> </tr></table>
+
\begin{equation*} \langle D \rangle = \sum _ { s } A ^ { T ( s ) } ( - A ^ { 2 } - A ^ { - 2 } ) ^ { | s D | - 1 },  \end{equation*}
  
where the sum is taken over all states of the link diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001016.png" /> and where a state codes the type of the smoothing performed at each crossing (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001018.png" /> type). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001019.png" /> denotes the number of smoothings of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001020.png" /> minus the number of smoothings of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001021.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001022.png" /> denotes the number of components of the diagram after all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001023.png" />-smoothings on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001024.png" /> are performed.
+
where the sum is taken over all states of the link diagram $D$ and where a state codes the type of the smoothing performed at each crossing ($L_0$ or $L _ { \infty }$ type). $T ( s )$ denotes the number of smoothings of type $L_0$ minus the number of smoothings of type $L _ { \infty }$. $| s D |$ denotes the number of components of the diagram after all $s$-smoothings on $D$ are performed.
  
The Kauffman bracket polynomial has a straightforward generalization to the solid torus (projected onto the annulus) and to the genus-two handlebody (projected onto the disc with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001025.png" /> holes). In the first case it has values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001026.png" /> and in the second case it has values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001027.png" />, see Fig.a2.
+
The Kauffman bracket polynomial has a straightforward generalization to the solid torus (projected onto the annulus) and to the genus-two handlebody (projected onto the disc with $2$ holes). In the first case it has values in $\mathbf{Z} [ A ^ { \pm 1 } , \alpha ]$ and in the second case it has values in $\mathbf{Z} [ A ^ { \pm 1 } , a , b , c ]$, see Fig.a2.
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k130010b.gif" />
+
<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/k130010b.gif" style="border:1px solid;"/>
  
 
Figure: k130010b
 
Figure: k130010b
  
For links in a solid torus the bracket polynomial can be used to estimate the wrapping number of the link. The wrapping conjecture says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001028.png" /> for a link diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001029.png" /> in the annulus is equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001030.png" />-degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001031.png" /> [[#References|[a1]]]. The Kauffman bracket skein module (cf. [[Skein module|Skein module]]) is a generalization of the Kauffman bracket polynomial to any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001032.png" />-dimensional manifold.
+
For links in a solid torus the bracket polynomial can be used to estimate the wrapping number of the link. The wrapping conjecture says that $\operatorname{wrap}( D )$ for a link diagram $D$ in the annulus is equal to the $\alpha$-degree of $\langle D \rangle$ [[#References|[a1]]]. The Kauffman bracket skein module (cf. [[Skein module|Skein module]]) is a generalization of the Kauffman bracket polynomial to any $3$-dimensional manifold.
  
The Kauffman bracket polynomial is a variant of the Jones polynomial. If one chooses an orientation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001033.png" /> on an unoriented link diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001034.png" />, then one defines an oriented link invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001036.png" /> is the Tait number (or writhe number) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001037.png" />, defined to be the sum of signs over all crossings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001038.png" />. Then the Jones polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001039.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001040.png" />. Furthermore, the Kauffman bracket polynomial satisfies:
+
The Kauffman bracket polynomial is a variant of the Jones polynomial. If one chooses an orientation $ \overset{\rightharpoonup}{ D }$ on an unoriented link diagram $D$, then one defines an oriented link invariant $f ( \overset{\rightharpoonup}{ D } ( A ) ) = ( - A ^ { 3 } ) ^ { - \operatorname { Tait } ( \overset{\rightharpoonup}{ D } ) } ( D )$, where $\operatorname{Tait}( \vec { D } )$ is the Tait number (or writhe number) of $ \overset{\rightharpoonup}{ D }$, defined to be the sum of signs over all crossings of $ \overset{\rightharpoonup}{ D }$. Then the Jones polynomial $V _ { L } ( t ) = f _ { L } ( A )$ for $t = A ^ { - 4 }$. Furthermore, the Kauffman bracket polynomial satisfies:
  
<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/k/k130/k130010/k13001041.png" /></td> </tr></table>
+
\begin{equation*} A \langle D _ { + } \rangle - A ^ { - 1 } \langle D _ { - } \rangle = ( A ^ { 2 } - A ^ { - 2 } ) \langle D _ { 0 } \rangle \end{equation*}
  
 
and
 
and
  
<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/k/k130/k130010/k13001042.png" /></td> </tr></table>
+
\begin{equation*} \langle D _ { + } \rangle + \langle D _ { - } \rangle = ( A + A ^ { - 1 } ) ( \langle D _ { 0 } \rangle + \langle D _ { \infty } \rangle ), \end{equation*}
  
 
and it is a specialization of both the [[Jones–Conway polynomial|Jones–Conway polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]].
 
and it is a specialization of both the [[Jones–Conway polynomial|Jones–Conway polynomial]] and the [[Kauffman polynomial|Kauffman polynomial]].
  
The Kauffman bracket polynomial was essential in the proof of the Tait conjectures on alternating links. In particular, for a connected alternating diagram without a nugatory crossing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001044.png" /> is the number of crossing points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001045.png" />. If the diagram is prime and non-alternating, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001046.png" />.
+
The Kauffman bracket polynomial was essential in the proof of the Tait conjectures on alternating links. In particular, for a connected alternating diagram without a nugatory crossing $\operatorname { span } \langle D \rangle = 4 c ( D )$, where $c ( D )$ is the number of crossing points of $D$. If the diagram is prime and non-alternating, then $\operatorname { span } \langle D \rangle < 4 c ( D )$.
  
The Kauffman bracket polynomial has several other applications, for example in the analysis of periodic links. The Reshetikhin–Turaev invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001047.png" />-manifolds can be constructed using the Kauffman bracket polynomial (via Kirby moves, cf. also [[Kirby calculus|Kirby calculus]]) [[#References|[a5]]].
+
The Kauffman bracket polynomial has several other applications, for example in the analysis of periodic links. The Reshetikhin–Turaev invariants of $3$-manifolds can be constructed using the Kauffman bracket polynomial (via Kirby moves, cf. also [[Kirby calculus|Kirby calculus]]) [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hoste,  J.H. Przytycki,  "An invariant of dichromatic links"  ''Proc. Amer. Math. Soc.'' , '''105''' :  4  (1989)  pp. 1003–1007</TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top">  L.H. Kauffman,  "State models and the Jones polynomial"  ''Topology'' , '''26'''  (1987)  pp. 395–407</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L.H. Kauffman,  "An invariant of regular isotopy"  ''Trans. Amer. Math. Soc.'' , '''318''' :  2  (1990)  pp. 417–471</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.B.R. Lickorish,  "An introduction to knot theory" , Springer  (1997)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W.M. Menasco,  M.B. Thistlethwaite,  "The classification of alternating links"  ''Ann. of Math.'' , '''138'''  (1993)  pp. 113–171</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  K. Murasugi,  "Jones polynomial and classical conjectures in knot theory"  ''Topology'' , '''26''' :  2  (1987)  pp. 187–194</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Traczyk,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001048.png" /> has no period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k130/k130010/k13001049.png" />: a criterion for periodic links"  ''Proc. Amer. Math. Soc.'' , '''180'''  (1990)  pp. 845–846</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M.B. Thistlethwaite,  "Kauffman polynomial and alternating links"  ''Topology'' , '''27'''  (1988)  pp. 311–318</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  J. Hoste,  J.H. Przytycki,  "An invariant of dichromatic links"  ''Proc. Amer. Math. Soc.'' , '''105''' :  4  (1989)  pp. 1003–1007</td></tr><tr><td valign="top">[a2]</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">[a3]</td> <td valign="top">  L.H. Kauffman,  "State models and the Jones polynomial"  ''Topology'' , '''26'''  (1987)  pp. 395–407</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L.H. Kauffman,  "An invariant of regular isotopy"  ''Trans. Amer. Math. Soc.'' , '''318''' :  2  (1990)  pp. 417–471</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  W.B.R. Lickorish,  "An introduction to knot theory" , Springer  (1997)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  W.M. Menasco,  M.B. Thistlethwaite,  "The classification of alternating links"  ''Ann. of Math.'' , '''138'''  (1993)  pp. 113–171</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  K. Murasugi,  "Jones polynomial and classical conjectures in knot theory"  ''Topology'' , '''26''' :  2  (1987)  pp. 187–194</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  P. Traczyk,  "$10_{101}$ has no period $7$: a criterion for periodic links"  ''Proc. Amer. Math. Soc.'' , '''180'''  (1990)  pp. 845–846</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M.B. Thistlethwaite,  "Kauffman polynomial and alternating links"  ''Topology'' , '''27'''  (1988)  pp. 311–318</td></tr>
 +
</table>
 +
 
 +
{{OldImage}}

Latest revision as of 07:42, 10 February 2024

An invariant of unoriented framed links.

It is a Laurent polynomial of one variable associated to ambient isotopy classes of unoriented framed links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed by L.H. Kauffman in the summer of 1985 and denoted by $\langle L \rangle$. It is defined recursively as follows: For a trivial link of $n$ components, with zero framing, $T _ { n }$, one puts

\begin{equation*} \langle T _ { n } \rangle = ( - A ^ { 2 } - A ^ { - 2 } ) ^ { n - 1 }. \end{equation*}

For the Kauffman bracket skein triple (cf. Fig.a1) one has the Kauffman bracket skein relation:

\begin{equation*} \langle L _ { + } \rangle = A \langle L _ { 0 } \rangle + A ^ { - 1 } \langle L _ { \infty } \rangle . \end{equation*}

Figure: k130010a

If $L ^ { ( 1 ) }$ is obtained from $L$ by a positive full twist on its framing, then $\langle L ^ { ( 1 ) } \rangle = - A ^ { 3 } \langle L \rangle$. The Kauffman bracket polynomial is also considered as an invariant of regular isotopy (Reidemeister moves: $R _ { 2 }$, $R _ { 3 }$, cf. Reidemeister theorem) of diagrams on the plane. $\langle D \rangle$ is changed by the first Reidemeister move by $- A ^ { \pm 3 }$.

The Kauffman bracket polynomial is related to a substitution of the dichromatic polynomial of signed graphs. This connection also relates the Kauffman bracket polynomial to the Potts model in statistical mechanics. State sum expansions of the dichromatic polynomial have their analogue for the Kauffman bracket polynomial. For example:

\begin{equation*} \langle D \rangle = \sum _ { s } A ^ { T ( s ) } ( - A ^ { 2 } - A ^ { - 2 } ) ^ { | s D | - 1 }, \end{equation*}

where the sum is taken over all states of the link diagram $D$ and where a state codes the type of the smoothing performed at each crossing ($L_0$ or $L _ { \infty }$ type). $T ( s )$ denotes the number of smoothings of type $L_0$ minus the number of smoothings of type $L _ { \infty }$. $| s D |$ denotes the number of components of the diagram after all $s$-smoothings on $D$ are performed.

The Kauffman bracket polynomial has a straightforward generalization to the solid torus (projected onto the annulus) and to the genus-two handlebody (projected onto the disc with $2$ holes). In the first case it has values in $\mathbf{Z} [ A ^ { \pm 1 } , \alpha ]$ and in the second case it has values in $\mathbf{Z} [ A ^ { \pm 1 } , a , b , c ]$, see Fig.a2.

Figure: k130010b

For links in a solid torus the bracket polynomial can be used to estimate the wrapping number of the link. The wrapping conjecture says that $\operatorname{wrap}( D )$ for a link diagram $D$ in the annulus is equal to the $\alpha$-degree of $\langle D \rangle$ [a1]. The Kauffman bracket skein module (cf. Skein module) is a generalization of the Kauffman bracket polynomial to any $3$-dimensional manifold.

The Kauffman bracket polynomial is a variant of the Jones polynomial. If one chooses an orientation $ \overset{\rightharpoonup}{ D }$ on an unoriented link diagram $D$, then one defines an oriented link invariant $f ( \overset{\rightharpoonup}{ D } ( A ) ) = ( - A ^ { 3 } ) ^ { - \operatorname { Tait } ( \overset{\rightharpoonup}{ D } ) } ( D )$, where $\operatorname{Tait}( \vec { D } )$ is the Tait number (or writhe number) of $ \overset{\rightharpoonup}{ D }$, defined to be the sum of signs over all crossings of $ \overset{\rightharpoonup}{ D }$. Then the Jones polynomial $V _ { L } ( t ) = f _ { L } ( A )$ for $t = A ^ { - 4 }$. Furthermore, the Kauffman bracket polynomial satisfies:

\begin{equation*} A \langle D _ { + } \rangle - A ^ { - 1 } \langle D _ { - } \rangle = ( A ^ { 2 } - A ^ { - 2 } ) \langle D _ { 0 } \rangle \end{equation*}

and

\begin{equation*} \langle D _ { + } \rangle + \langle D _ { - } \rangle = ( A + A ^ { - 1 } ) ( \langle D _ { 0 } \rangle + \langle D _ { \infty } \rangle ), \end{equation*}

and it is a specialization of both the Jones–Conway polynomial and the Kauffman polynomial.

The Kauffman bracket polynomial was essential in the proof of the Tait conjectures on alternating links. In particular, for a connected alternating diagram without a nugatory crossing $\operatorname { span } \langle D \rangle = 4 c ( D )$, where $c ( D )$ is the number of crossing points of $D$. If the diagram is prime and non-alternating, then $\operatorname { span } \langle D \rangle < 4 c ( D )$.

The Kauffman bracket polynomial has several other applications, for example in the analysis of periodic links. The Reshetikhin–Turaev invariants of $3$-manifolds can be constructed using the Kauffman bracket polynomial (via Kirby moves, cf. also Kirby calculus) [a5].

References

[a1] J. Hoste, J.H. Przytycki, "An invariant of dichromatic links" Proc. Amer. Math. Soc. , 105 : 4 (1989) pp. 1003–1007
[a2] V.F.R. Jones, "Hecke algebra representations of braid groups and link polynomials" Ann. of Math. , 126 : 2 (1987) pp. 335–388
[a3] L.H. Kauffman, "State models and the Jones polynomial" Topology , 26 (1987) pp. 395–407
[a4] L.H. Kauffman, "An invariant of regular isotopy" Trans. Amer. Math. Soc. , 318 : 2 (1990) pp. 417–471
[a5] W.B.R. Lickorish, "An introduction to knot theory" , Springer (1997)
[a6] W.M. Menasco, M.B. Thistlethwaite, "The classification of alternating links" Ann. of Math. , 138 (1993) pp. 113–171
[a7] K. Murasugi, "Jones polynomial and classical conjectures in knot theory" Topology , 26 : 2 (1987) pp. 187–194
[a8] P. Traczyk, "$10_{101}$ has no period $7$: a criterion for periodic links" Proc. Amer. Math. Soc. , 180 (1990) pp. 845–846
[a9] M.B. Thistlethwaite, "Kauffman polynomial and alternating links" Topology , 27 (1988) pp. 311–318


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How to Cite This Entry:
Kauffman bracket polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kauffman_bracket_polynomial&oldid=12747
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article