Khovanov homology for links in \(\#^r (S^2\times S^1)\)
type: Item
instance of: scholarly article
title: Khovanov homology for links in \(\#^r (S^2\times S^1)\)
zbMATH Open document ID: 1496.57016
author: Michael Willis
published in: Michigan Mathematical Journal
publication date: 2022-02-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1812.06584
review text: In this paper the author generalises Khovanov homology to links in a \(3\)-sphere with handles attached. When \(r=0\), the manifold \(M^r = S^3\) and the author recovers classical Khovanov homology. When \(r=1\), \(M^r = S^2\times S^1\) and the author recovers the Khovanov homology for links in \(S^2\times S^1\) of \textit{L. Rozansky} [``A categorification of the stable SU(2) Witten-Reshetikhin-Turaev invariant of links in \(S^2\times S^1\)'', Preprint, \url{arXiv:1011.1958}]. The \(3\)-sphere with \(r\) handles \(M^r\) can be visualised as the complement in \(S^3\) of \(r\) pairs of balls, where the boundary of each ball is identified with the boundary of the other ball in its pair. Now fix a segment in the boundary of each ball, so that the two segments in each pair of balls are identified. Fix also an embedding of a square \([0,1]^2\) in \(S^3\) (without removing the balls) that isotopes each segment into its paired segment. A link in \(M^r\) can be represented as a tangle in that complement disjoint from the embedded square whose intersection with the boundary of each ball lies in the segment and has the same cardinality as the intersection with the boundary of the other ball in the pair. To each such representation of a link \(L\) one can associate a sequence of links \(\{L(k)_{k\geq 0}\}\) in \(S^3\) as follows: \(L(0)\) is the result of closing the tangle by gluing parallel strands in each square. \(L(k)\) is the result of performing \(k\) full twists to the parallel strands in each square of \(L(0)\). After appropriate shiftings and homotopy replacements whose definition takes two fifths of the paper, the Khovanov chain complexes of the links \(\{L(k)\}_{k\geq 0}\) form a directed system and the Khovanov chain complex of the link \(L\) is its colimit. The Khovanov homology of the link \(L\) is the homology of this colimit, which is shown to be a link invariant and functorial with respect to cobordisms up to sign. These constructions apply only if the inters…
reviewed by: Federico Cantero
Mathematics Subject Classification ID: 57K18
zbMATH DE Number: 7473043
zbMATH Keywords: Khovanov homology
zbMATH Keywords: skein module
zbMATH Keywords: WRT invariants
zbMATH Keywords: Jones-Wenzl projectors
MaRDI profile type: Publication
OpenAlex ID: W3040816620
arXiv ID: 1812.06584
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DOI: 10.1307/MMJ/1594281620
description: scientific article