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An invariant of oriented links (cf. also [[Knot theory|Knot theory]]).
 
An invariant of oriented links (cf. also [[Knot theory|Knot theory]]).
  
It is a Laurent polynomial of two variables associated to ambient isotopy classes of links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200401.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200402.png" />), constructed by L. Kauffman in 1985 and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200403.png" /> (cf. also [[Isotopy|Isotopy]]).
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It is a Laurent polynomial of two variables associated to ambient isotopy classes of links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed by L. Kauffman in 1985 and denoted by $F _ { L } ( a , x )$ (cf. also [[Isotopy|Isotopy]]).
  
The construction starts from the invariant of non-oriented link diagrams (cf. also [[Knot and link diagrams|Knot and link diagrams]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200404.png" />. For a diagram of a trivial link of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200405.png" /> components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200406.png" />, put
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The construction starts from the invariant of non-oriented link diagrams (cf. also [[Knot and link diagrams|Knot and link diagrams]]), $\Lambda _ { D } ( a , x )$. For a diagram of a trivial link of $n$ components, $T _ { n }$, put
  
<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/k120/k120040/k1200407.png" /></td> </tr></table>
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\begin{equation*} \Lambda _ { T _ { n } } ( a , x ) = \left( \frac { a + a ^ { - 1 } - x } { x } \right) ^ { n - 1 }. \end{equation*}
  
 
The Kauffman skein quadruple satisfies a skein relation
 
The Kauffman skein quadruple satisfies a 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/k120/k120040/k1200408.png" /></td> </tr></table>
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\begin{equation*} \Lambda _ { D _ { + } } ( a , x ) + \Lambda _ { D _ { - } } ( a , x ) = x ( \Lambda _ { D _ { 0 } } ( a , x ) + \Lambda _ { D _ { \infty } } ( a , x ) ). \end{equation*}
  
Furthermore, the second and the third Reidemeister moves (cf. also [[Knot and link diagrams|Knot and link diagrams]]) preserve the invariant, while the first Reidemeister move is changing it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k1200409.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004010.png" /> (depending on whether the move is positive or negative). To define the Kauffman polynomial from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004011.png" /> one considers an oriented link diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004012.png" />, represented by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004013.png" />, and puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004015.png" /> is the Tait (or writhe) number of an oriented link diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004016.png" /> (cf. also [[Writhing number|Writhing number]]).
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Furthermore, the second and the third Reidemeister moves (cf. also [[Knot and link diagrams|Knot and link diagrams]]) preserve the invariant, while the first Reidemeister move is changing it by $a$ or $a ^ { - 1 }$ (depending on whether the move is positive or negative). To define the Kauffman polynomial from $\Lambda _ { L } ( a , x )$ one considers an oriented link diagram $L _ { D }$, represented by $D$, and puts $F _ { L _ { D } } ( a , x ) = a ^ { - \text { Tait } ( L _ { D } ) } \Lambda _ { D } ( a , x )$, where $\operatorname{Tait}( L _ { D } )$ is the Tait (or writhe) number of an oriented link diagram $L _ { D }$ (cf. also [[Writhing number|Writhing number]]).
  
The Jones polynomial and its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004017.png" />-cable version are special cases of the Kauffman polynomial.
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The Jones polynomial and its $2$-cable version are special cases of the Kauffman polynomial.
  
There are several examples of different links with the same Kauffman polynomial. In particular, the knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004018.png" /> and its mirror image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004019.png" /> are different but have the same Kauffman polynomial. Some other examples deal with mutant and their generalizations: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004020.png" />-rotor constructions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004021.png" />-cables of mutants and any satellites of connected sums (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004023.png" />). The following two questions are open (1998) and of great interest:
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There are several examples of different links with the same Kauffman polynomial. In particular, the knot $9_{42}$ and its mirror image $\overline { 9 } _ { 42 }$ are different but have the same Kauffman polynomial. Some other examples deal with mutant and their generalizations: $3$-rotor constructions, $2$-cables of mutants and any satellites of connected sums ($K _ { 1 } \# K _ { 2 }$ and $K _ { 1 } \# - K _ { 2 }$). The following two questions are open (1998) and of great interest:
  
1) Is there a non-trivial knot with Kauffman polynomial equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004024.png" />?
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1) Is there a non-trivial knot with Kauffman polynomial equal to $1$?
  
 
2) Is there infinite number of different knots with the same Kauffman polynomial?
 
2) Is there infinite number of different knots with the same Kauffman polynomial?
  
The number of Fox <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004026.png" />-colourings can be computed from the Kauffman polynomial (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004028.png" />). Kauffman constructed his polynomial building on his interpretation of the Jones–Conway (HOMFLYpt), the Brandt–Lickorish–Millett and the Ho polynomials.
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The number of Fox $3$-colourings can be computed from the Kauffman polynomial (at $a = 1$, $x = - 1$). Kauffman constructed his polynomial building on his interpretation of the Jones–Conway (HOMFLYpt), the Brandt–Lickorish–Millett and the Ho polynomials.
  
The important feature of the Kauffman polynomial is its computational complexity (with respect to the number of crossings of the diagram; cf. also [[Complexity theory|Complexity theory]]). It is conjectured to be exponential and it is proven to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004030.png" />-hard (cf. also [[NP|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004031.png" />]]); so, up to the conjecture <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004032.png" />, the Kauffman polynomial cannot be computed in polynomial time. The complexity of computing the Alexander polynomial is, in contrast, polynomial. The Kauffman polynomial is independent from the Alexander polynomial, it often distinguishes a knot from its mirror image but, for example, it does not distinguish the knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004034.png" /> (in Perko's notation), but the Alexander polynomial does distinguish these knots. The Kauffman polynomials are stratified by the Vassiliev invariants, which have polynomial-time computational complexity.
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The important feature of the Kauffman polynomial is its computational complexity (with respect to the number of crossings of the diagram; cf. also [[Complexity theory|Complexity theory]]). It is conjectured to be exponential and it is proven to be $\cal N P$-hard (cf. also [[NP|$\cal N P$]]); so, up to the conjecture $\mathcal{N P} \neq \mathcal{P}$, the Kauffman polynomial cannot be computed in polynomial time. The complexity of computing the Alexander polynomial is, in contrast, polynomial. The Kauffman polynomial is independent from the Alexander polynomial, it often distinguishes a knot from its mirror image but, for example, it does not distinguish the knots $11_{255}$ and $11_{257}$ (in Perko's notation), but the Alexander polynomial does distinguish these knots. The Kauffman polynomials are stratified by the Vassiliev invariants, which have polynomial-time computational complexity.
  
If one considers the skein relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004035.png" />, one gets the Dubrovnik polynomial, which is a variant of the Kauffman polynomial.
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If one considers the skein relation $\Lambda _ { D _ { + } } ^ { * } ( a , x ) - \Lambda _ { D _ { - } } ^ { * } ( a , x ) = x ( \Lambda _ { D _ { 0 } } ^ { * } ( a , x ) - \Lambda _ { D _ { \infty } } ^ { * } ( a , x ) )$, one gets the Dubrovnik polynomial, which is a variant of the Kauffman polynomial.
  
The Kauffman polynomial leads to the Kauffman skein module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004036.png" />-manifolds.
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The Kauffman polynomial leads to the Kauffman skein module of $3$-manifolds.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Anstee,  J.H. Przytycki,  D. Rolfsen,  "Knot polynomials and generalized mutation"  ''Topol. Appl.'' , '''32'''  (1989)  pp. 237–249</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.D. Brandt,  W.B.R. Lickorish,  K.C. Millett,  "A polynomial invariant for unoriented knots and links"  ''Invent. Math.'' , '''84'''  (1986)  pp. 563–573</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.R. Garey,  D.S. Johnson,  "Computers and intractability: A guide to theory of NP completeness" , Freeman  (1979)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C.F. Ho,  "A new polynomial for knots and links; preliminary report"  ''Abstracts Amer. Math. Soc.'' , '''6''' :  4  (1985)  pp. 300</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Hoste,  J.H. Przytycki,  "A survey of skein modules of 3-manifolds"  A. Kawauchi (ed.) , ''Knots 90, Proc. Internat. Conf. Knot Theory and Related Topics, Osaka (Japan, August 15-19, 1990 )'' , W. de Gruyter  (1992)  pp. 363–379</TD></TR><TR><TD valign="top">[a6]</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">[a7]</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">[a8]</TD> <TD valign="top">  W.B.R. Lickorish,  "Polynomials for links"  ''Bull. London Math. Soc.'' , '''20'''  (1988)  pp. 558–588</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  W.B.R. Lickorish,  K.C. Millett,  "An evaluation of the F-polynomial of a link" , ''Differential topology: Proc. 2nd Topology Symp., Siegen / FRG 1987'' , ''Lecture Notes Math.'' , '''1350'''  (1988)  pp. 104–108</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J.H. Przytycki,  "Equivalence of cables of mutants of knots"  ''Canad. J. Math.'' , '''XLI''' :  2  (1989)  pp. 250–273</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J.H. Przytycki,  "Skein modules of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120040/k12004037.png" />-manifolds"  ''Bull. Acad. Polon. Math.'' , '''39''' :  1–2  (1991)  pp. 91–100</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M.B. Thistlethwaite,  "On the Kauffman polynomial of an adequate link"  ''Invent. Math.'' , '''93'''  (1988)  pp. 285–296</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.G. Turaev,  "The Conway and Kauffman modules of the solid torus"  ''J. Soviet Math.'' , '''52''' :  1  (1990)  pp. 2799–2805  ''Zap. Nauchn. Sem. Lomi'' , '''167'''  (1988)  pp. 79–89</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  R.P. Anstee,  J.H. Przytycki,  D. Rolfsen,  "Knot polynomials and generalized mutation"  ''Topol. Appl.'' , '''32'''  (1989)  pp. 237–249</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.D. Brandt,  W.B.R. Lickorish,  K.C. Millett,  "A polynomial invariant for unoriented knots and links"  ''Invent. Math.'' , '''84'''  (1986)  pp. 563–573</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M.R. Garey,  D.S. Johnson,  "Computers and intractability: A guide to theory of NP completeness" , Freeman  (1979)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C.F. Ho,  "A new polynomial for knots and links; preliminary report"  ''Abstracts Amer. Math. Soc.'' , '''6''' :  4  (1985)  pp. 300</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J. Hoste,  J.H. Przytycki,  "A survey of skein modules of 3-manifolds"  A. Kawauchi (ed.) , ''Knots 90, Proc. Internat. Conf. Knot Theory and Related Topics, Osaka (Japan, August 15-19, 1990 )'' , W. de Gruyter  (1992)  pp. 363–379</td></tr><tr><td valign="top">[a6]</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">[a7]</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">[a8]</td> <td valign="top">  W.B.R. Lickorish,  "Polynomials for links"  ''Bull. London Math. Soc.'' , '''20'''  (1988)  pp. 558–588</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  W.B.R. Lickorish,  K.C. Millett,  "An evaluation of the F-polynomial of a link" , ''Differential topology: Proc. 2nd Topology Symp., Siegen / FRG 1987'' , ''Lecture Notes Math.'' , '''1350'''  (1988)  pp. 104–108</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J.H. Przytycki,  "Equivalence of cables of mutants of knots"  ''Canad. J. Math.'' , '''XLI''' :  2  (1989)  pp. 250–273</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J.H. Przytycki,  "Skein modules of $3$-manifolds"  ''Bull. Acad. Polon. Math.'' , '''39''' :  1–2  (1991)  pp. 91–100</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  M.B. Thistlethwaite,  "On the Kauffman polynomial of an adequate link"  ''Invent. Math.'' , '''93'''  (1988)  pp. 285–296</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  V.G. Turaev,  "The Conway and Kauffman modules of the solid torus"  ''J. Soviet Math.'' , '''52''' :  1  (1990)  pp. 2799–2805  ''Zap. Nauchn. Sem. Lomi'' , '''167'''  (1988)  pp. 79–89</td></tr></table>

Latest revision as of 17:02, 1 July 2020

An invariant of oriented links (cf. also Knot theory).

It is a Laurent polynomial of two variables associated to ambient isotopy classes of links in $\mathbf{R} ^ { 3 }$ (or $S ^ { 3 }$), constructed by L. Kauffman in 1985 and denoted by $F _ { L } ( a , x )$ (cf. also Isotopy).

The construction starts from the invariant of non-oriented link diagrams (cf. also Knot and link diagrams), $\Lambda _ { D } ( a , x )$. For a diagram of a trivial link of $n$ components, $T _ { n }$, put

\begin{equation*} \Lambda _ { T _ { n } } ( a , x ) = \left( \frac { a + a ^ { - 1 } - x } { x } \right) ^ { n - 1 }. \end{equation*}

The Kauffman skein quadruple satisfies a skein relation

\begin{equation*} \Lambda _ { D _ { + } } ( a , x ) + \Lambda _ { D _ { - } } ( a , x ) = x ( \Lambda _ { D _ { 0 } } ( a , x ) + \Lambda _ { D _ { \infty } } ( a , x ) ). \end{equation*}

Furthermore, the second and the third Reidemeister moves (cf. also Knot and link diagrams) preserve the invariant, while the first Reidemeister move is changing it by $a$ or $a ^ { - 1 }$ (depending on whether the move is positive or negative). To define the Kauffman polynomial from $\Lambda _ { L } ( a , x )$ one considers an oriented link diagram $L _ { D }$, represented by $D$, and puts $F _ { L _ { D } } ( a , x ) = a ^ { - \text { Tait } ( L _ { D } ) } \Lambda _ { D } ( a , x )$, where $\operatorname{Tait}( L _ { D } )$ is the Tait (or writhe) number of an oriented link diagram $L _ { D }$ (cf. also Writhing number).

The Jones polynomial and its $2$-cable version are special cases of the Kauffman polynomial.

There are several examples of different links with the same Kauffman polynomial. In particular, the knot $9_{42}$ and its mirror image $\overline { 9 } _ { 42 }$ are different but have the same Kauffman polynomial. Some other examples deal with mutant and their generalizations: $3$-rotor constructions, $2$-cables of mutants and any satellites of connected sums ($K _ { 1 } \# K _ { 2 }$ and $K _ { 1 } \# - K _ { 2 }$). The following two questions are open (1998) and of great interest:

1) Is there a non-trivial knot with Kauffman polynomial equal to $1$?

2) Is there infinite number of different knots with the same Kauffman polynomial?

The number of Fox $3$-colourings can be computed from the Kauffman polynomial (at $a = 1$, $x = - 1$). Kauffman constructed his polynomial building on his interpretation of the Jones–Conway (HOMFLYpt), the Brandt–Lickorish–Millett and the Ho polynomials.

The important feature of the Kauffman polynomial is its computational complexity (with respect to the number of crossings of the diagram; cf. also Complexity theory). It is conjectured to be exponential and it is proven to be $\cal N P$-hard (cf. also $\cal N P$); so, up to the conjecture $\mathcal{N P} \neq \mathcal{P}$, the Kauffman polynomial cannot be computed in polynomial time. The complexity of computing the Alexander polynomial is, in contrast, polynomial. The Kauffman polynomial is independent from the Alexander polynomial, it often distinguishes a knot from its mirror image but, for example, it does not distinguish the knots $11_{255}$ and $11_{257}$ (in Perko's notation), but the Alexander polynomial does distinguish these knots. The Kauffman polynomials are stratified by the Vassiliev invariants, which have polynomial-time computational complexity.

If one considers the skein relation $\Lambda _ { D _ { + } } ^ { * } ( a , x ) - \Lambda _ { D _ { - } } ^ { * } ( a , x ) = x ( \Lambda _ { D _ { 0 } } ^ { * } ( a , x ) - \Lambda _ { D _ { \infty } } ^ { * } ( a , x ) )$, one gets the Dubrovnik polynomial, which is a variant of the Kauffman polynomial.

The Kauffman polynomial leads to the Kauffman skein module of $3$-manifolds.

References

[a1] R.P. Anstee, J.H. Przytycki, D. Rolfsen, "Knot polynomials and generalized mutation" Topol. Appl. , 32 (1989) pp. 237–249
[a2] R.D. Brandt, W.B.R. Lickorish, K.C. Millett, "A polynomial invariant for unoriented knots and links" Invent. Math. , 84 (1986) pp. 563–573
[a3] M.R. Garey, D.S. Johnson, "Computers and intractability: A guide to theory of NP completeness" , Freeman (1979)
[a4] C.F. Ho, "A new polynomial for knots and links; preliminary report" Abstracts Amer. Math. Soc. , 6 : 4 (1985) pp. 300
[a5] J. Hoste, J.H. Przytycki, "A survey of skein modules of 3-manifolds" A. Kawauchi (ed.) , Knots 90, Proc. Internat. Conf. Knot Theory and Related Topics, Osaka (Japan, August 15-19, 1990 ) , W. de Gruyter (1992) pp. 363–379
[a6] 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
[a7] L.H. Kauffman, "An invariant of regular isotopy" Trans. Amer. Math. Soc. , 318 : 2 (1990) pp. 417–471
[a8] W.B.R. Lickorish, "Polynomials for links" Bull. London Math. Soc. , 20 (1988) pp. 558–588
[a9] W.B.R. Lickorish, K.C. Millett, "An evaluation of the F-polynomial of a link" , Differential topology: Proc. 2nd Topology Symp., Siegen / FRG 1987 , Lecture Notes Math. , 1350 (1988) pp. 104–108
[a10] J.H. Przytycki, "Equivalence of cables of mutants of knots" Canad. J. Math. , XLI : 2 (1989) pp. 250–273
[a11] J.H. Przytycki, "Skein modules of $3$-manifolds" Bull. Acad. Polon. Math. , 39 : 1–2 (1991) pp. 91–100
[a12] M.B. Thistlethwaite, "On the Kauffman polynomial of an adequate link" Invent. Math. , 93 (1988) pp. 285–296
[a13] V.G. Turaev, "The Conway and Kauffman modules of the solid torus" J. Soviet Math. , 52 : 1 (1990) pp. 2799–2805 Zap. Nauchn. Sem. Lomi , 167 (1988) pp. 79–89
How to Cite This Entry:
Kauffman polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kauffman_polynomial&oldid=18266
This article was adapted from an original article by J. Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article