Introduction to Knot Theory

Crowell, Richard H.; Fox, Ralph H.

doi:10.1007/978-1-4612-9935-6

Cited by 212 publications

(203 citation statements)

References 4 publications

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“…The boundary operators d ρ 1 and d ρ 2 are induced by the usual ones of C * ( M K , Z) obtained using the Fox differential calculus (see [CF,Chap. 7…”

Section: The Knot Exterior Twisted Chain Complexmentioning

confidence: 99%

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Dubois¹

2005

Annales De L’institut Fourier

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For a knot K in $S^3$ and a regular representation $ ho$ of its group $G_K$ into SU(2) we construct a non abelian Reidemeister torsion on the first twisted cohomology group of the knot exterior. This non abelian Reidemeister torsion provides a volume form on the SU(2)-representation space of $G_K$. In another way, we construct according to Casson--or more precisely taking into account Lin's and Heusener's further works--a volume form on the SU(2)-representation space of $G_K$. Next, we compare these two apparently different points of view--the first by means of the Reidemeister torsion and the second defined ``a la Casson"--and finally prove that they define the same topological knot invariant

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“…The boundary operators d ρ 1 and d ρ 2 are induced by the usual ones of C * ( M K , Z) obtained using the Fox differential calculus (see [CF,Chap. 7…”

Section: The Knot Exterior Twisted Chain Complexmentioning

confidence: 99%

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Dubois¹

2005

Annales De L’institut Fourier

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“…This lemma follows directly from the proof of Theorem 5. The rest of Theorem 6 follows from the well-known fact that a connected sum of non-trivial knots is non-trivial; see, for example, [3]. 4.1.…”

Section: The Underlying Genus Of Virtual Knotsmentioning

confidence: 99%

Untitled

2008

Trans. Moscow Math. Soc.

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Abstract. The theory of virtual knots is a generalization of the theory of classical knots proposed by Kauffman in 1996. In this paper solutions of certain problems of the theory of virtual knots are given. The first part of the paper is devoted to the representation of virtual knots as knots in 3-manifolds of the form S g × I modulo stabilization. Using Haken's theory of normal surfaces, a theorem is proved showing that the problem of recognizing virtual links is algorithmically soluble. A theorem is proved stating that (any) connected sum of non-trivial virtual knots is non-trivial. This result is a corollary of inequalities proved in this paper for the (underlying) genus of virtual knots. The last part of this paper is concerned with long virtual knots. By contrast with the classical case, the classifications of long and compact virtual knots are different. Invariants of long virtual knots are constructed with the help of which one can establish the inequivalence of long knots having equivalent closures and also that some pairs of long virtual knots do not commute.

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“…Below we consider less trivial examples. here ∂ ∂b is an analog of the Fox' derivative [7] adapted to semigroups in [18].…”

Section: Partial Cohomologiesmentioning

confidence: 99%

Semigroup Cohomology and Applications

Novikov¹

2001

Algebra — Representation Theory

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This article is a survey of the author's research. It consists of three sections concerned three kinds of cohomologies of semigroups. Section 1 considers 'classic' cohomology as it was introduced by Eilenberg and MacLane. Here the attention is concentrated mainly on semigroups having cohomological dimension 1. In Section 2 a generalization of the Eilenberg-MacLane cohomology is introduced, the so-called 0-cohomology, which appears in applied topics (projective representations of semigroups, Brauer monoids). At last Section 3 is devoted to further generalizing: partial cohomology defined and discussed in it are used then for calculation of the classic cohomology for some semigroups.I am indebted to Prof. K. Roggenkamp and Prof. M. Ştefanescu for the support of my participation in the Workshop.

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Introduction to Knot Theory

Cited by 212 publications

References 4 publications

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Non abelian Reidemeister torsion and volume form on the SU(2)-representation space of knot groups

Untitled

Semigroup Cohomology and Applications

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