Skein module of links in a handlebody

Topology '90 ◽  
2011 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Skein Module

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

scholarly journals The Yang–Mills measure in the SU(3)-skein module

2017 ◽  
Vol 26 (11) ◽  
pp. 1750071
Author(s):  
Charles Frohman ◽  
Jianyuan K. Zhong
Keyword(s):  
Complex Number ◽  
Positive Real Number ◽  
Closed Formula ◽  
Positive Real ◽  
Local Diffeomorphism ◽  
Yang Mills ◽  
Skein Module ◽  
Root Of Unity ◽  
Trivalent Graphs ◽  
Math Phys

Let [Formula: see text] be a nonzero complex number which is not a root of unity. Let [Formula: see text] be a compact oriented surface, the [Formula: see text]-skein space of [Formula: see text], [Formula: see text], is the vector space over [Formula: see text] generated by framed oriented links (including framed oriented trivalent graphs in [Formula: see text]) quotient by the [Formula: see text]-skein relations due to Kuperberg [Spiders for rank [Formula: see text] Lie algebra, Comm. Math. Phys. 180(1) (1996) 109–151]. For closed [Formula: see text], with genus greater than [Formula: see text], we construct a local diffeomorphism invariant trace on [Formula: see text] when [Formula: see text] is a positive real number not equal to [Formula: see text].

On the Homflypt skein module of $S^1 \times S^2$

2001 ◽  
Vol 237 (4) ◽  
pp. 769-814 ◽  
Author(s):  
Patrick M. Gilmer ◽  
Jianyuan K. Zhong
Keyword(s):  
Skein Module

The Kauffman bracket skein module ofS 1×S 2

1995 ◽  
Vol 220 (1) ◽  
pp. 65-73 ◽  
Author(s):  
Jim Hoste ◽  
Józef H. Przytycki
Keyword(s):  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

THE CHARACTER VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

2012 ◽  
Vol 23 (01) ◽  
pp. 1250015 ◽  
Author(s):  
KHALED QAZAQZEH
Keyword(s):  
Character Variety ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
One Relator Groups

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link Lp/q of two components modulo the (A + 1)-torsion.

THE (2, ∞)-SKEIN MODULE OF LENS SPACES: A GENERALIZATION OF THE JONES POLYNOMIAL

1993 ◽  
Vol 02 (03) ◽  
pp. 321-333 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Lens Spaces ◽  
Skein Module

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.

THE (2, ∞)-SKEIN MODULE OF WHITEHEAD MANIFOLDS

1995 ◽  
Vol 04 (03) ◽  
pp. 411-427 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Skein Modules ◽  
Skein Module ◽  
Natural Filtration ◽  
Stark Contrast ◽  
Torsion Free

To any orientable 3-manifold one can associate a module, called the (2, ∞)-skein module, which is essentially a generalization of the Jones polynomial of links in S3. For an uncountable collection of open contractible 3-manifolds, each constructed in a fashion similar to the classic Whitehead manifold, we prove that their (2, ∞)-skein modules are infinitely generated, torsion free, but not free. These examples stand in stark contrast to [Formula: see text], whose (2, ∞)-skein module is free on one generator. To each of these manifolds we associate a subgroup G of the rationals which may be interpreted via wrapping numbers. We show that the skein module of M has a natural filtration by modules indexed by G. For the specific case of the Whitehead manifold, we describe its (2, ∞)-skein module and associated filtration in greater detail.

scholarly journals Non-commutative trigonometry and the A-polynomial of the trefoil knot

2002 ◽  
Vol 133 (2) ◽  
pp. 311-323 ◽  
Author(s):  
RĂZVAN GELCA
Keyword(s):  
Character Variety ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Trefoil Knot ◽  
Sum Formula ◽  
Left And Right ◽  
Trefoil Knots ◽  
The Relationship ◽  
The Ideal

The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character variety of the fundamental group, which was explained in [2]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.

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