Floer homology and singular knots

Ozsváth, Peter; Stipsicz, András I.; Szabó, Zoltán

doi:10.1112/jtopol/jtp015

Cited by 28 publications

(52 citation statements)

References 12 publications

(31 reference statements)

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“…The Alexander polynomials of a cube of resolutions (in Vassiliev's sense) of a singular knot were categorified in [1]. Moreover, a 1-variable extension of the Alexander polynomial for singular links was categorified in [11]. The generalized cube of resolutions (containing Vassilievs resolutions as well as those smoothings at double points which preserve the orientation) was categorified in [12].…”

Section: Introduction and Resultsmentioning

confidence: 99%

The Jones and Alexander Polynomials for Singular Links

Fiedler

2010

J. Knot Theory Ramifications

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Section: Introduction and Resultsmentioning

confidence: 99%

The Jones and Alexander Polynomials for Singular Links

Fiedler

2010

J. Knot Theory Ramifications

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“…Remark 6.8. It is interesting to compare the 'figure-8' curve to the local Heegaard diagram for a singular crossing in [23], as the number of generators agree for the second loop, up to an additional tensor factor. Also note that the proposition above gives rise to an exact triangle similar to the one in [30].…”

Section: Skein Exact Sequencesmentioning

confidence: 99%

Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

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“…The knot Floer edge ring is defined over Z[t −1 , t]], the ring of Laurent series in t, while the HOMFLY-PT edge ring is defined over Q [8,13] or Z [9]. The variable t appears in the definition of the knot Floer ideals as well because the knot Floer graph homology is in fact the singular knot Floer homology [11] of the graph in S 3 , computed with a particular choice of twisted coefficients. (2) The non-local ideal.…”

Section: • •mentioning

confidence: 99%

“…Ozsváth and Szabó constructed the original cube of resolutions for knot Floer homology from the collection of B(G e I ) for I ∈ {0, 1} n , which arose for them as singular knot Floer homology [11] with twisted coefficients. They showed that their cube of resolutions complex is the E 1 page of a spectral sequence to HF K − (K) that collapses at the E 2 page [12, Theorem 1.1, Section 5].…”

Section: If That Conjecture Holds It Would Follow Immediately That Bmentioning

confidence: 99%

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

Gilmore¹

2015

Algebr. Geom. Topol.

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Abstract. This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen's conjecture that a spectral sequence relates the two knot homologies. We replace the language of twisted coefficients with the more quantum topological language of framings on trivalent graphs. We define a homology theory for framed trivalent graphs with boundary that -for a particular non-blackboard framing -specializes to the homology of singular knots underlying the knot Floer cube of resolutions. For blackboard framed graphs, our theory conjecturally recovers the graph homology underlying the HOMFLY-PT chain complex. We explain the appearance of the non-local ideal by expressing it as an ideal quotient of an ideal that appears in both the HOMFLY-PT and knot Floer cubes of resolutions. This result is a corollary of our main theorem, which is that closing a strand in a braid graph corresponds to taking an ideal quotient of its non-local ideal. The proof is a Gröbner basis argument that connects the combinatorics of the non-local ideal to those of Buchberger's Algorithm.

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Floer homology and singular knots

Cited by 28 publications

References 12 publications

The Jones and Alexander Polynomials for Singular Links

The Jones and Alexander Polynomials for Singular Links

Peculiar modules for 4‐ended tangles

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

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