A combinatorial spanning tree model for knot Floer homology
type: Item
instance of: scholarly article
title: A combinatorial spanning tree model for knot Floer homology
zbMATH Open document ID: 1262.57013
author: John A. Baldwin
author: Adam Simon Levine
published in: Advances in Mathematics
publication date: 2012-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.5199
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6094102
zbMATH Keywords: knot Floer homology
zbMATH Keywords: Heegaard Floer homology
zbMATH Keywords: Khovanov homology
zbMATH Keywords: spanning tree
zbMATH Keywords: exact triangle
reviewed by: Elisenda Grigsby
describes a project that uses: HFK
MaRDI profile type: Publication
OpenAlex ID: W2019321194
arXiv ID: 1105.5199
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DOI: 10.1016/J.AIM.2012.06.006
review text: Knot Floer homology [\textit{P. Ozsváth} and \textit{Z. Szabó}, Adv. Math. 186, No. 1, 58--116 (2004; Zbl 1062.57019) and \textit{J. Rasmussen}, Floer homology and Knot complements. PhD Thesis. Harvard U. (2003)] is an invariant of oriented links in \(S^3\) taking the form of a bigraded abelian group realized as the homology of an abstract chain complex. It categorifies the Alexander polynomial, detects the knot genus, fiberedness, bounds the slice genus, and provides obstructions to the existence of certain exceptional Dehn surgeries. Although the original definition involved counts of holomorphic disks in certain symplectic manifolds, purely combinatorial descriptions have since been found [\textit{C. Manolescu}, \textit{P. Ozsváth} and \textit{S. Sarkar}, Ann. Math. (2) 169, No. 2, 633--660 (2009; Zbl 1179.57022) and \textit{S. Sarkar} and \textit{J. Wang}, ibid. (2) 171, No. 2, 1213--1236 (2010; Zbl 1228.57017)].NEWLINENEWLINEIn the present work, the authors give a new combinatorial description of a grading-collapsed (\(\delta\)--graded) version of knot Floer homology with coefficients in the field, \(\mathcal{F} = \mathbb{F}(T)\), of rational functions in a single variable over the field \(\mathbb{F} = \mathbb{Z}/2\mathbb{Z}\). The construction takes as its starting point a skein exact triangle for knot Floer homology introduced in [\textit{C. Manolescu}, Math. Res. Lett. 14, No. 5--6, 839--852 (2007; Zbl 1228.57017)]. This yields a combinatorial ``cube of resolutions'' description of the \(\delta\)--graded knot Floer homology reminiscent of the ``twisted'' spanning tree models of \(\delta\)--graded Khovanov homology described in [\textit{L. Roberts}, Totally twisted Khovanov homology (2011), \url{arXiv:1109.0508}] and [\textit{T. Jaeger}, A remark on Roberts' totally twisted Khovanov homology (2011), \url{arXiv:1109.1805}]. Recall that Khovanov homology [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] is another invariant of or…
description: scientific article; zbMATH DE number 6094102