A combinatorial spanning tree model for knot Floer homology
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 oriented links in \(S^3\) that categorifies the Jones polynomial.NEWLINENEWLINEIndeed, the authors speculate that a good understanding of the relationship between their spanning tree model and that of Roberts and Jaeger should lead to an explicit understanding of the connection between Khovanov homology and knot Floer homology, an understanding that has so far proved elusive (cf. [\textit{J. Rasmussen}, Fields Inst. Comm. 47, 261--280 (2005; Zbl 1095.57016)]).
scientific article; zbMATH DE number 6094102
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