Skein module

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