View source for Skein module

''linear skein''

An algebraic object associated to a [[Manifold|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|module]] associated to a [[Three-dimensional manifold|three-dimensional manifold]] by considering linear combinations of links in the manifold, modulo properly chosen (skein) relations (cf. also [[Link|Link]]; [[Linear skein|Linear skein]]). It is the main object of [[Algebraic topology based on knots|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 <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" />)

<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>

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.

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]]).

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.

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]]].

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" />.

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]]].

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]]).

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:

<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>

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:

<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>

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]]].

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]]).

====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>