Algebraic & Geometric Topology

A Jones polynomial for braid-like isotopies of oriented links and its categorification

Benjamin Audoux and Thomas Fiedler

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Abstract

A braid-like isotopy for links in 3–space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 4 (2005), 1535-1553.

Dates
Received: 9 March 2005
Revised: 20 October 2005
Accepted: 24 October 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796488

Digital Object Identifier
doi:10.2140/agt.2005.5.1535

Mathematical Reviews number (MathSciNet)
MR2186108

Zentralblatt MATH identifier
1084.57010

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 20F36: Braid groups; Artin groups

Keywords
braid-like isotopies Jones polynomials Khovanov homologies

Citation

Audoux, Benjamin; Fiedler, Thomas. A Jones polynomial for braid-like isotopies of oriented links and its categorification. Algebr. Geom. Topol. 5 (2005), no. 4, 1535--1553. doi:10.2140/agt.2005.5.1535. https://projecteuclid.org/euclid.agt/1513796488


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References

  • M M Asaeda, J H Przytycki, A S Sikora, Categorification of the Kauffman bracket skein module of $I$-bundles over surfaces, Algebr. Geom. Topol. 4 (2004) 1177–1210
  • B Audoux, Homologie de Khovanov, available from: http://doctorants.picard.ups-tlse.fr/doks/gdte/audoux_khovanov.pdf
  • J S Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J. (1974)
  • G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. Berlin (1985)
  • T Fiedler, A small state sum for knots, Topology 32 (1993) 281–294
  • T Fiedler, Gauss diagram invariants for knots and links, Mathematics and its Applications 532, Kluwer Academic Publishers, Dordrecht (2001)
  • J Hoste, J H Przytycki, An invariant of dichromatic links, Proc. Amer. Math. Soc. 105 (1989) 1003–1007
  • L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407
  • M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359–426
  • H R Morton, Infinitely many fibred knots having the same Alexander polynomial, Topology 17 (1978) 101–104
  • O Viro, Remarks on definition of Khovanov homology, e-print (2002) \arxivmath.GT/0202199