A spanning tree cohomology theory for links
type: Item
instance of: scholarly article
title: A spanning tree cohomology theory for links
zbMATH Open document ID: 1309.57009
author: Daniel Kriz
author: Igor Kriz
published in: Advances in Mathematics
publication date: 2014-04-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.0064
review text: There are many new knot invariants including the Khovanov homology, which is a sequence of homology groups whose Euler characteristic is the Jones polynomial, the knot Floer homology of Ozsváth and Szabó, which is similarly related to the Alexander polynomial, and the Heegaard-Floer homology of the branched double cover along a link. The Heegaard-Floer homology was introduced by \textit{P. Ozsváth} and \textit{Z. Szabó} [Duke Math. J. 129, No. 1, 39--61 (2005; Zbl 1083.57042)] as a link invariant of the branched double cover \(\Sigma(L)\) of \(S^3\) along a link \(L\), and they discovered a spectral sequence from Khovanov homology to \(\widehat{HF}(\Sigma(L))\) such that every \(E_r\)-term of this sequence is a link invariant. Recently, \textit{J. Baldwin}, \textit{P. Ozsváth}, and \textit{Z. Szabó} [Heegaard Floer homology of double-covers, Kauffman states, and Novikov rings, to appear] have introduced a perturbed Heegaard-Floer homology with coefficients in a Novikov ring, constructed an appropriate BOS-spectral sequence, and showed that the \(E_2\)-term is trivial and the cochain complex \((E_2,d_2)\) has a combinatorial description. The main purpose of this paper is to show that the \(E_3\)-term of the BOS-spectral sequence is an invariant of oriented links. Also, the authors present concrete examples of computation of this invariant. If \(L\) is an oriented link in \(S^3\) with generic projection \({\mathcal D}\) such that every connected component of \(S^3\smallsetminus{\mathcal D}\) is simply connected, then \((C({\mathcal D}),\Phi)\) is the cochain complex which is the \(E_2\)-term of the BOS-spectral sequence converging to the Heegaard-Floer homology \(\widehat{HF}(\sum(L))\), where \(\Phi(T)=\sum\limits_{T'}\left(\frac{1}{1+\alpha(T,T')}+\frac{1}{1+\beta(T,T')}\right)T'\) for a spanning tree \(T\). They prove that if \(F\) is a field of characteristic \(2\) with elements \(u_f\), \(z_v\), then for each \(i\), \((*)\,\text{rank}_F\left(H^i(C({\mathcal D},F,…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6284978
zbMATH Keywords: spanning tree
zbMATH Keywords: invariants of knots
zbMATH Keywords: cohomology theory
zbMATH Keywords: links
MaRDI profile type: Publication
OpenAlex ID: W2962779069
arXiv ID: 1109.0064
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DOI: 10.1016/J.AIM.2014.01.006
description: scientific article