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''linear skein''
 
''linear skein''
  
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iii) Does the module reflect the topology/geometry of a three-dimensional manifold (e.g. surfaces in a manifold, geometric decomposition of a manifold)?
 
iii) Does the module reflect the topology/geometry of a three-dimensional manifold (e.g. surfaces in a manifold, geometric decomposition of a manifold)?
  
iv) Does the module admit some additional structure (e.g. filtration, gradation, multiplication, Hopf algebra structure)? One of the simplest skein modules is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303401.png" />-deformation of the first [[Homology group|homology group]] of a three-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303402.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303403.png" />. It is based on the skein relation (between non-oriented framed links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303404.png" />)
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iv) Does the module admit some additional structure (e.g. filtration, gradation, multiplication, Hopf algebra structure)? One of the simplest skein modules is a $q$-deformation of the first [[Homology group|homology group]] of a three-dimensional manifold $M$, denoted by $\mathcal{S} _ { 2 } ( M ; q )$. It is based on the skein relation (between non-oriented framed links in $M$)
  
<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/s/s130/s130340/s1303405.png" /></td> </tr></table>
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\begin{equation*} L _ { + } = q L _ { 0 }. \end{equation*}
  
Already this simply defined skein module  "sees"  non-separating surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303406.png" />. These surfaces are responsible for the torsion part of this skein module.
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Already this simply defined skein module  "sees"  non-separating surfaces in $M$. These surfaces are responsible for the torsion part of this skein module.
  
 
There is a more general pattern: most of the skein modules analyzed reflect various surfaces in a manifold.
 
There is a more general pattern: most of the skein modules analyzed reflect various surfaces in a manifold.
  
The best studied skein modules use skein relations which worked successfully in classical knot theory (when defining polynomial invariants of links in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303407.png" />, cf. also [[Link|Link]]).
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The best studied skein modules use skein relations which worked successfully in classical knot theory (when defining polynomial invariants of links in $\mathbf{R} ^ { 3 }$, cf. also [[Link|Link]]).
  
1) The Kauffman bracket skein module is based on the Kauffman bracket skein relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303408.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s1303409.png" />. Among the Jones-type skein modules it is the one best understood. It can be interpreted as a quantization of the coordinate ring of the character variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034010.png" /> representations of the [[Fundamental group|fundamental group]] of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034011.png" />, [[#References|[a4]]], [[#References|[a2]]], [[#References|[a16]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034012.png" />, the Kauffman bracket skein module is an [[Algebra|algebra]] (usually non-commutative). It is a finitely-generated algebra for a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034013.png" /> [[#References|[a3]]], and has no zero divisors [[#References|[a16]]]. Incompressible tori and two-dimensional spheres in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034014.png" /> yield torsion in the Kauffman bracket skein module; it is a question of fundamental importance whether other surfaces can yield torsion as well.
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1) The Kauffman bracket skein module is based on the Kauffman bracket skein relation $L _ { + } = A L _ { - } + A ^ { - 1 } L _ { \infty }$, and is denoted by $S _ { 2 , \infty} ( M )$. Among the Jones-type skein modules it is the one best understood. It can be interpreted as a quantization of the coordinate ring of the character variety of $\operatorname{SL} ( 2 , \mathbf{C} )$ representations of the [[Fundamental group|fundamental group]] of the manifold $M$, [[#References|[a4]]], [[#References|[a2]]], [[#References|[a16]]]. For $M = F \times [ 0,1 ]$, the Kauffman bracket skein module is an [[Algebra|algebra]] (usually non-commutative). It is a finitely-generated algebra for a compact $F$ [[#References|[a3]]], and has no zero divisors [[#References|[a16]]]. Incompressible tori and two-dimensional spheres in $M$ yield torsion in the Kauffman bracket skein module; it is a question of fundamental importance whether other surfaces can yield torsion as well.
  
2) Skein modules based on the Jones–Conway relation (Homflypt relation) are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034015.png" /> and generalize skein modules based on the Conway relation which were hinted at by J.H. Conway. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034017.png" /> is a [[Hopf algebra|Hopf algebra]] (usually neither commutative nor co-commutative), [[#References|[a19]]], [[#References|[a11]]]. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034018.png" /> is a free module and can be interpreted as a quantization [[#References|[a6]]], [[#References|[a18]]], [[#References|[a10]]], [[#References|[a19]]] (cf. also [[Drinfel'd–Turaev quantization|Drinfel'd–Turaev quantization]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034019.png" /> is related to the algebraic set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034020.png" /> representations of the fundamental group of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034021.png" />, [[#References|[a17]]].
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2) Skein modules based on the Jones–Conway relation (Homflypt relation) are denoted by $S _ { 3 } ( M )$ and generalize skein modules based on the Conway relation which were hinted at by J.H. Conway. For $M = F \times [ 0,1 ]$, $S _ { 3 } ( M )$ is a [[Hopf algebra|Hopf algebra]] (usually neither commutative nor co-commutative), [[#References|[a19]]], [[#References|[a11]]]. $S _ { 3 } ( F \times [ 0,1 ] )$ is a free module and can be interpreted as a quantization [[#References|[a6]]], [[#References|[a18]]], [[#References|[a10]]], [[#References|[a19]]] (cf. also [[Drinfel'd–Turaev quantization|Drinfel'd–Turaev quantization]]). $S _ { 3 } ( M )$ is related to the algebraic set of $\operatorname{SL} ( n , \mathbf{C} )$ representations of the fundamental group of the manifold $M$, [[#References|[a17]]].
  
3) The skein module based on the Kauffman polynomial relation is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034022.png" /> and is known to be free for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034023.png" />.
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3) The skein module based on the Kauffman polynomial relation is denoted by $S _ { 3 , \infty }$ and is known to be free for $M = F \times [ 0,1 ]$.
  
4) In homotopy skein modules, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034024.png" /> for self-crossings. The best studied example is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034026.png" />-homotopy skein module with the skein relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034027.png" /> for mixed crossings. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034028.png" /> it is a quantization, [[#References|[a7]]], [[#References|[a19]]], [[#References|[a15]]], and as noted by U. Kaiser they can be almost completely understood using Lin's singular tori technique [[#References|[a20]]].
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4) In homotopy skein modules, $L _ { + } = L _ { - }$ for self-crossings. The best studied example is the $q$-homotopy skein module with the skein relation $q ^ { - 1 } L _ { + } - q L _ { - } = z L _ { 0 }$ for mixed crossings. For $M = F \times [ 0,1 ]$ it is a quantization, [[#References|[a7]]], [[#References|[a19]]], [[#References|[a15]]], and as noted by U. Kaiser they can be almost completely understood using Lin's singular tori technique [[#References|[a20]]].
  
5) The only studied skein module based on relations deforming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034030.png" />-moves to date (2000) is the fourth skein module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034031.png" />, with possible additional framing relation. It is conjectured that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034032.png" /> this module is generated by trivial links. Motivation for this is the Montesinos–Nakanishi three-move conjecture (cf. [[Montesinos–Nakanishi conjecture|Montesinos–Nakanishi conjecture]]).
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5) The only studied skein module based on relations deforming $n$-moves to date (2000) is the fourth skein module $\mathcal{S} _ { 4 } ( M ) = R \mathcal{L} / ( b _ { 0 } L _ { 0 } + b _ { 1 } L _ { 1 } + b _ { 2 } L _ { 2 } + b _ { 3 } L _ { 3 } )$, with possible additional framing relation. It is conjectured that in $S ^ { 3 }$ this module is generated by trivial links. Motivation for this is the Montesinos–Nakanishi three-move conjecture (cf. [[Montesinos–Nakanishi conjecture|Montesinos–Nakanishi conjecture]]).
  
6) Extending the family of knots, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034033.png" />, by singular knots, and resolving singular crossing by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034034.png" /> allows one to define the Vassiliev–Gusarov filtration:
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6) Extending the family of knots, $\mathcal{K}$, by singular knots, and resolving singular crossing by $K _ { cr } = K _ { + } - K _ { - }$ allows one to define the Vassiliev–Gusarov filtration:
  
<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/s/s130/s130340/s13034035.png" /></td> </tr></table>
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\begin{equation*} \ldots \subset C _ { 3 } \subset \ldots \subset C _ { 2 } \subset \ldots \subset C _ { 1 } \subset \ldots \subset C _ { 0 } = R \cal K \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034036.png" /> is generated by knots with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034037.png" /> singular points. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034038.png" />th Vassiliev–Gusarov skein module is defined to be a quotient:
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where $C _ { k }$ is generated by knots with $k$ singular points. The $k$th Vassiliev–Gusarov skein module is defined to be a quotient:
  
<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/s/s130/s130340/s13034039.png" /></td> </tr></table>
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\begin{equation*} W _ { k } ( M ) = R \mathcal{K} / C _ { k + 1 }. \end{equation*}
  
The completion of the space of knots with respect to the Vassiliev–Gusarov filtration, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034040.png" />, is a [[Hopf algebra|Hopf algebra]] (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034041.png" />). Functions dual to Vassiliev–Gusarov skein modules are called finite type or Vassiliev invariants of knots, [[#References|[a12]]].
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The completion of the space of knots with respect to the Vassiliev–Gusarov filtration, $\widehat { R \mathcal{K} }$, is a [[Hopf algebra|Hopf algebra]] (for $M = S ^ { 3 }$). Functions dual to Vassiliev–Gusarov skein modules are called finite type or Vassiliev invariants of knots, [[#References|[a12]]].
  
Skein modules have their origin in the observation by J.W. Alexander [[#References|[a1]]] that his polynomials of three links, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034044.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034045.png" />, are linearly related. They were envisioned by Conway (linear skein) [[#References|[a5]]] and the outline of the theory was given first in the spring of 1987 [[#References|[a9]]] after Jones' construction of his polynomial (the Jones polynomial) in 1984; see [[#References|[a8]]], [[#References|[a13]]], [[#References|[a14]]] for the history of the development of skein modules. V.G. Turaev pointed out the importance of skein modules as quantizations, [[#References|[a19]]] (cf. also [[Drinfel'd–Turaev quantization|Drinfel'd–Turaev quantization]]).
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Skein modules have their origin in the observation by J.W. Alexander [[#References|[a1]]] that his polynomials of three links, $L _ { + }$, $L_{-}$ and $L_0$ in $\mathbf{R} ^ { 3 }$, are linearly related. They were envisioned by Conway (linear skein) [[#References|[a5]]] and the outline of the theory was given first in the spring of 1987 [[#References|[a9]]] after Jones' construction of his polynomial (the Jones polynomial) in 1984; see [[#References|[a8]]], [[#References|[a13]]], [[#References|[a14]]] for the history of the development of skein modules. V.G. Turaev pointed out the importance of skein modules as quantizations, [[#References|[a19]]] (cf. also [[Drinfel'd–Turaev quantization|Drinfel'd–Turaev quantization]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Alexander,  "Topological invariants of knots and links"  ''Trans. Amer. Math. Soc.'' , '''30'''  (1928)  pp. 275–306</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D. Bullock,  C. Frohman,  J. Kania–Bartoszyńska,  "Understanding the Kauffman bracket skein module"  ''J. Knot Th. Ramifications''  (1999)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Bullock,  "A finite set of generators for the Kauffman bracket skein algebra"  ''Math. Z.'' , '''231''' :  1  (1999)  pp. 91–101</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Bullock,  "Rings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034046.png" />-characters and the Kauffman bracket skein module"  ''Comment. Math. Helv.'' , '''72'''  (1997)  pp. 521–542</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon  (1969)  pp. 329–358</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Hoste,  M. Kidwell,  "Dichromatic link invariants"  ''Trans. Amer. Math. Soc.'' , '''321''' :  1  (1990)  pp. 197–229</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J. Hoste,  J.H. Przytycki,  "Homotopy skein modules of oriented 3-manifolds"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 475–488</TD></TR><TR><TD valign="top">[a8]</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)'' , de Gruyter  (1992)  pp. 363–379</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J.H. Przytycki,  "Skein modules of 3-manifolds"  ''Bull. Polish Acad. Sci.'' , '''39''' :  1–2  (1991)  pp. 91–100</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J.H. Przytycki,  "Skein module of links in a handlebody"  B. Apanasov (ed.)  W.D. Neumann (ed.)  A.W. Reid (ed.)  L. Siebenmann (ed.) , ''Topology 90, Proc. Research Sem. Low Dimensional Topology at OSU'' , de Gruyter  (1992)  pp. 315–342</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J.H. Przytycki,  "Quantum group of links in a handlebody"  M. Gerstenhaber (ed.)  J.D. Stasheff (ed.) , ''Contemporary Math.: Deformation Theory and Quantum Groups with Applications to Mathematical Physics'' , '''134'''  (1992)  pp. 235–245</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.H. Przytycki,  "Vassiliev–Gusarov skein modules of 3-manifolds and criteria for periodicity of knots"  K. Johannson (ed.) , ''Low-Dimensional Topology (Knoxville, 1992)'' , Internat. Press, Cambridge, Mass.  (1994)  pp. 157–176</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  J.H. Przytycki,  "Algebraic topology based on knots: an introduction"  S. Suzuki (ed.) , ''Knots 96, Proc. Fifth Internat. Research Inst. MSJ'' , World Sci.  (1997)  pp. 279–297</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J.H. Przytycki,  "Fundamentals of Kauffman bracket skein modules"  ''Kobe Math. J.'' , '''16''' :  1  (1999)  pp. 45–66</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  J.H. Przytycki,  "Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs" , ''Proc. Conf. Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, New York, March, 14-15, 1998)''  (2001)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  J.H. Przytycki,  A.S. Sikora,  "On skein algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034047.png" />-character varieties"  ''Topology'' , '''39''' :  1  (2000)  pp. 115–148</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  A.S. Sikora,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130340/s13034048.png" />-character varieties as spaces of graphs"  ''Trans. Amer. Math. Soc.'' , '''353'''  (2001)  pp. 2773–2804</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  V.G. Turaev,  "The Conway and Kauffman modules of the solid torus"  ''J. Soviet Math.'' , '''52'''  (1990)  pp. 2799–2805  ''Zap. Nauchn. Sem. LOMI'' , '''167'''  (1988)  pp. 79–89</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  V.G. Turaev,  "Skein quantization of Poisson algebras of loops on surfaces"  ''Ann. Sci. École Norm. Sup.'' , '''4''' :  24  (1991)  pp. 635–704</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  V. Kaiser,  "Presentations of homotopy skein modules of oriented 3-manifolds"  ''J. Knot Th. Ramifications'' , '''10''' :  3  (2001)  pp. 461–491</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  J.W. Alexander,  "Topological invariants of knots and links"  ''Trans. Amer. Math. Soc.'' , '''30'''  (1928)  pp. 275–306</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  D. Bullock,  C. Frohman,  J. Kania–Bartoszyńska,  "Understanding the Kauffman bracket skein module"  ''J. Knot Th. Ramifications''  (1999)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Bullock,  "A finite set of generators for the Kauffman bracket skein algebra"  ''Math. Z.'' , '''231''' :  1  (1999)  pp. 91–101</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D. Bullock,  "Rings of $S l _ { 2 } ( C )$-characters and the Kauffman bracket skein module"  ''Comment. Math. Helv.'' , '''72'''  (1997)  pp. 521–542</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  J.H. Conway,  "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon  (1969)  pp. 329–358</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J. Hoste,  M. Kidwell,  "Dichromatic link invariants"  ''Trans. Amer. Math. Soc.'' , '''321''' :  1  (1990)  pp. 197–229</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J. Hoste,  J.H. Przytycki,  "Homotopy skein modules of oriented 3-manifolds"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 475–488</td></tr><tr><td valign="top">[a8]</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)'' , de Gruyter  (1992)  pp. 363–379</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J.H. Przytycki,  "Skein modules of 3-manifolds"  ''Bull. Polish Acad. Sci.'' , '''39''' :  1–2  (1991)  pp. 91–100</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J.H. Przytycki,  "Skein module of links in a handlebody"  B. Apanasov (ed.)  W.D. Neumann (ed.)  A.W. Reid (ed.)  L. Siebenmann (ed.) , ''Topology 90, Proc. Research Sem. Low Dimensional Topology at OSU'' , de Gruyter  (1992)  pp. 315–342</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  J.H. Przytycki,  "Quantum group of links in a handlebody"  M. Gerstenhaber (ed.)  J.D. Stasheff (ed.) , ''Contemporary Math.: Deformation Theory and Quantum Groups with Applications to Mathematical Physics'' , '''134'''  (1992)  pp. 235–245</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  J.H. Przytycki,  "Vassiliev–Gusarov skein modules of 3-manifolds and criteria for periodicity of knots"  K. Johannson (ed.) , ''Low-Dimensional Topology (Knoxville, 1992)'' , Internat. Press, Cambridge, Mass.  (1994)  pp. 157–176</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  J.H. Przytycki,  "Algebraic topology based on knots: an introduction"  S. Suzuki (ed.) , ''Knots 96, Proc. Fifth Internat. Research Inst. MSJ'' , World Sci.  (1997)  pp. 279–297</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  J.H. Przytycki,  "Fundamentals of Kauffman bracket skein modules"  ''Kobe Math. J.'' , '''16''' :  1  (1999)  pp. 45–66</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  J.H. Przytycki,  "Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs" , ''Proc. Conf. Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, New York, March, 14-15, 1998)''  (2001)</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  J.H. Przytycki,  A.S. Sikora,  "On skein algebras and $S l _ { 2 } ( C )$-character varieties"  ''Topology'' , '''39''' :  1  (2000)  pp. 115–148</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  A.S. Sikora,  "$P S L_n$-character varieties as spaces of graphs"  ''Trans. Amer. Math. Soc.'' , '''353'''  (2001)  pp. 2773–2804</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  V.G. Turaev,  "The Conway and Kauffman modules of the solid torus"  ''J. Soviet Math.'' , '''52'''  (1990)  pp. 2799–2805  ''Zap. Nauchn. Sem. LOMI'' , '''167'''  (1988)  pp. 79–89</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  V.G. Turaev,  "Skein quantization of Poisson algebras of loops on surfaces"  ''Ann. Sci. École Norm. Sup.'' , '''4''' :  24  (1991)  pp. 635–704</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  V. Kaiser,  "Presentations of homotopy skein modules of oriented 3-manifolds"  ''J. Knot Th. Ramifications'' , '''10''' :  3  (2001)  pp. 461–491</td></tr></table>

Latest revision as of 16:45, 1 July 2020

linear skein

An algebraic object associated to a manifold, usually constructed as a formal linear combination of embedded (or immersed) submanifolds, modulo locally defined relations. In a more restricted setting, a skein module is a module associated to a three-dimensional manifold by considering linear combinations of links in the manifold, modulo properly chosen (skein) relations (cf. also Link; Linear skein). It is the main object of algebraic topology based on knots. In the choice of relations one takes into account several factors:

i) Is the module obtained accessible (computable)?

ii) How precise are the modules in distinguishing three-dimensional manifolds and links in them?

iii) Does the module reflect the topology/geometry of a three-dimensional manifold (e.g. surfaces in a manifold, geometric decomposition of a manifold)?

iv) Does the module admit some additional structure (e.g. filtration, gradation, multiplication, Hopf algebra structure)? One of the simplest skein modules is a $q$-deformation of the first homology group of a three-dimensional manifold $M$, denoted by $\mathcal{S} _ { 2 } ( M ; q )$. It is based on the skein relation (between non-oriented framed links in $M$)

\begin{equation*} L _ { + } = q L _ { 0 }. \end{equation*}

Already this simply defined skein module "sees" non-separating surfaces in $M$. These surfaces are responsible for the torsion part of this skein module.

There is a more general pattern: most of the skein modules analyzed reflect various surfaces in a manifold.

The best studied skein modules use skein relations which worked successfully in classical knot theory (when defining polynomial invariants of links in $\mathbf{R} ^ { 3 }$, cf. also Link).

1) The Kauffman bracket skein module is based on the Kauffman bracket skein relation $L _ { + } = A L _ { - } + A ^ { - 1 } L _ { \infty }$, and is denoted by $S _ { 2 , \infty} ( M )$. Among the Jones-type skein modules it is the one best understood. It can be interpreted as a quantization of the coordinate ring of the character variety of $\operatorname{SL} ( 2 , \mathbf{C} )$ representations of the fundamental group of the manifold $M$, [a4], [a2], [a16]. For $M = F \times [ 0,1 ]$, the Kauffman bracket skein module is an algebra (usually non-commutative). It is a finitely-generated algebra for a compact $F$ [a3], and has no zero divisors [a16]. Incompressible tori and two-dimensional spheres in $M$ yield torsion in the Kauffman bracket skein module; it is a question of fundamental importance whether other surfaces can yield torsion as well.

2) Skein modules based on the Jones–Conway relation (Homflypt relation) are denoted by $S _ { 3 } ( M )$ and generalize skein modules based on the Conway relation which were hinted at by J.H. Conway. For $M = F \times [ 0,1 ]$, $S _ { 3 } ( M )$ is a Hopf algebra (usually neither commutative nor co-commutative), [a19], [a11]. $S _ { 3 } ( F \times [ 0,1 ] )$ is a free module and can be interpreted as a quantization [a6], [a18], [a10], [a19] (cf. also Drinfel'd–Turaev quantization). $S _ { 3 } ( M )$ is related to the algebraic set of $\operatorname{SL} ( n , \mathbf{C} )$ representations of the fundamental group of the manifold $M$, [a17].

3) The skein module based on the Kauffman polynomial relation is denoted by $S _ { 3 , \infty }$ and is known to be free for $M = F \times [ 0,1 ]$.

4) In homotopy skein modules, $L _ { + } = L _ { - }$ for self-crossings. The best studied example is the $q$-homotopy skein module with the skein relation $q ^ { - 1 } L _ { + } - q L _ { - } = z L _ { 0 }$ for mixed crossings. For $M = F \times [ 0,1 ]$ it is a quantization, [a7], [a19], [a15], and as noted by U. Kaiser they can be almost completely understood using Lin's singular tori technique [a20].

5) The only studied skein module based on relations deforming $n$-moves to date (2000) is the fourth skein module $\mathcal{S} _ { 4 } ( M ) = R \mathcal{L} / ( b _ { 0 } L _ { 0 } + b _ { 1 } L _ { 1 } + b _ { 2 } L _ { 2 } + b _ { 3 } L _ { 3 } )$, with possible additional framing relation. It is conjectured that in $S ^ { 3 }$ this module is generated by trivial links. Motivation for this is the Montesinos–Nakanishi three-move conjecture (cf. Montesinos–Nakanishi conjecture).

6) Extending the family of knots, $\mathcal{K}$, by singular knots, and resolving singular crossing by $K _ { cr } = K _ { + } - K _ { - }$ allows one to define the Vassiliev–Gusarov filtration:

\begin{equation*} \ldots \subset C _ { 3 } \subset \ldots \subset C _ { 2 } \subset \ldots \subset C _ { 1 } \subset \ldots \subset C _ { 0 } = R \cal K \end{equation*}

where $C _ { k }$ is generated by knots with $k$ singular points. The $k$th Vassiliev–Gusarov skein module is defined to be a quotient:

\begin{equation*} W _ { k } ( M ) = R \mathcal{K} / C _ { k + 1 }. \end{equation*}

The completion of the space of knots with respect to the Vassiliev–Gusarov filtration, $\widehat { R \mathcal{K} }$, is a Hopf algebra (for $M = S ^ { 3 }$). Functions dual to Vassiliev–Gusarov skein modules are called finite type or Vassiliev invariants of knots, [a12].

Skein modules have their origin in the observation by J.W. Alexander [a1] that his polynomials of three links, $L _ { + }$, $L_{-}$ and $L_0$ in $\mathbf{R} ^ { 3 }$, are linearly related. They were envisioned by Conway (linear skein) [a5] and the outline of the theory was given first in the spring of 1987 [a9] after Jones' construction of his polynomial (the Jones polynomial) in 1984; see [a8], [a13], [a14] for the history of the development of skein modules. V.G. Turaev pointed out the importance of skein modules as quantizations, [a19] (cf. also Drinfel'd–Turaev quantization).

References

[a1] J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306
[a2] D. Bullock, C. Frohman, J. Kania–Bartoszyńska, "Understanding the Kauffman bracket skein module" J. Knot Th. Ramifications (1999)
[a3] D. Bullock, "A finite set of generators for the Kauffman bracket skein algebra" Math. Z. , 231 : 1 (1999) pp. 91–101
[a4] D. Bullock, "Rings of $S l _ { 2 } ( C )$-characters and the Kauffman bracket skein module" Comment. Math. Helv. , 72 (1997) pp. 521–542
[a5] J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon (1969) pp. 329–358
[a6] J. Hoste, M. Kidwell, "Dichromatic link invariants" Trans. Amer. Math. Soc. , 321 : 1 (1990) pp. 197–229
[a7] J. Hoste, J.H. Przytycki, "Homotopy skein modules of oriented 3-manifolds" Math. Proc. Cambridge Philos. Soc. , 108 (1990) pp. 475–488
[a8] 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) , de Gruyter (1992) pp. 363–379
[a9] J.H. Przytycki, "Skein modules of 3-manifolds" Bull. Polish Acad. Sci. , 39 : 1–2 (1991) pp. 91–100
[a10] J.H. Przytycki, "Skein module of links in a handlebody" B. Apanasov (ed.) W.D. Neumann (ed.) A.W. Reid (ed.) L. Siebenmann (ed.) , Topology 90, Proc. Research Sem. Low Dimensional Topology at OSU , de Gruyter (1992) pp. 315–342
[a11] J.H. Przytycki, "Quantum group of links in a handlebody" M. Gerstenhaber (ed.) J.D. Stasheff (ed.) , Contemporary Math.: Deformation Theory and Quantum Groups with Applications to Mathematical Physics , 134 (1992) pp. 235–245
[a12] J.H. Przytycki, "Vassiliev–Gusarov skein modules of 3-manifolds and criteria for periodicity of knots" K. Johannson (ed.) , Low-Dimensional Topology (Knoxville, 1992) , Internat. Press, Cambridge, Mass. (1994) pp. 157–176
[a13] J.H. Przytycki, "Algebraic topology based on knots: an introduction" S. Suzuki (ed.) , Knots 96, Proc. Fifth Internat. Research Inst. MSJ , World Sci. (1997) pp. 279–297
[a14] J.H. Przytycki, "Fundamentals of Kauffman bracket skein modules" Kobe Math. J. , 16 : 1 (1999) pp. 45–66
[a15] J.H. Przytycki, "Homotopy and q-homotopy skein modules of 3-manifolds: An example in Algebra Situs" , Proc. Conf. Low-Dimensional Topology in Honor of Joan Birman's 70th Birthday (Columbia Univ./Barnard College, New York, March, 14-15, 1998) (2001)
[a16] J.H. Przytycki, A.S. Sikora, "On skein algebras and $S l _ { 2 } ( C )$-character varieties" Topology , 39 : 1 (2000) pp. 115–148
[a17] A.S. Sikora, "$P S L_n$-character varieties as spaces of graphs" Trans. Amer. Math. Soc. , 353 (2001) pp. 2773–2804
[a18] V.G. Turaev, "The Conway and Kauffman modules of the solid torus" J. Soviet Math. , 52 (1990) pp. 2799–2805 Zap. Nauchn. Sem. LOMI , 167 (1988) pp. 79–89
[a19] V.G. Turaev, "Skein quantization of Poisson algebras of loops on surfaces" Ann. Sci. École Norm. Sup. , 4 : 24 (1991) pp. 635–704
[a20] V. Kaiser, "Presentations of homotopy skein modules of oriented 3-manifolds" J. Knot Th. Ramifications , 10 : 3 (2001) pp. 461–491
How to Cite This Entry:
Skein module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skein_module&oldid=16183
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article