Bordered Sutured Floer Homology

Zarev, Rumen

doi:10.7916/d83r10v4

Cited by 5 publications

(8 citation statements)

References 19 publications

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“…Nonetheless, we do have a glueing formula for peculiar modules CFT ∂ (T ). Its proof relies on Zarev's Glueing Theorem for his bordered sutured invariants [40] and an identification of some structure maps of certain bordered sutured invariants for tangles and peculiar modules. The precise statement of our Glueing Theorem uses the -tensor product between type A and type D structures familiar from bordered Heegaard Floer homology; for details, see Definition 1.18.…”

Section: Peculiar Modulesmentioning

confidence: 99%

“…We will assume some familiarity with [38][39][40] and only give a short review of the basic geometric objects involved.…”

Section: Review Of Bordered Sutured Heegaard Floer Theorymentioning

confidence: 99%

“…

With a 4-ended tangle T , we associate a Heegaard Floer invariant CFT ∂ (T ), the peculiar module of T . Based on Zarev's bordered sutured Heegaard Floer theory [40], we prove a glueing formula for this invariant which recovers link Floer homology HFL. Moreover, we classify peculiar modules in terms of immersed curves on the 4-punctured sphere.

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confidence: 98%

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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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Section: Peculiar Modulesmentioning

confidence: 99%

“…We will assume some familiarity with [38][39][40] and only give a short review of the basic geometric objects involved.…”

Section: Review Of Bordered Sutured Heegaard Floer Theorymentioning

confidence: 99%

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Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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“…In general, HFT can be upgraded to a glueable theory by modifying its differential. This approach is described in my PhD thesis [Zib16, section II.3] and uses more complicated arc diagrams from Zarev's bordered sutured theory [Zar11]. For 4-ended tangles, one can define a slightly different glueing structure, which turns out to be very similar to Hanselman, J. Rasmussen and Watson's immersed curve invariant for 3-manifolds with torus boundary [HRW16].…”

Section: Towards δ -Graded Mutation Invariance Of Hflmentioning

confidence: 99%

“…Actually, we define HFT for tangles within arbitrary 3-manifolds M with spherical boundary, see the comment at the beginning of section 4. This is done using Zarev's bordered sutured Heegaard Floer theory [Zar11]. However, in general, the gradings on his invariants are rather complicated.…”

Section: Introductionmentioning

confidence: 99%

Kauffman states and Heegaard diagrams for tangles

Zibrowius

2019

Algebr. Geom. Topol.

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We define polynomial tangle invariants ∇ s T via Kauffman states and Alexander codes and investigate some of their properties. In particular, we prove symmetry relations for ∇ s T of 4-ended tangles and deduce that the multivariable Alexander polynomial is invariant under Conway mutation. The invariants ∇ s T can be interpreted naturally via Heegaard diagrams for tangles. This leads to a categorified version of ∇ s T : a Heegaard Floer homology HFT for tangles, which we define as a bordered sutured invariant. We discuss a bigrading on HFT and prove symmetry relations for HFT of 4-ended tangles that echo those for ∇ s T .Kauffman states and Heegaard diagrams for tangles 3 of L and we call R the mutating tangle in this mutation. If L is oriented, we choose an orientation of L that agrees with the one for L outside of R. If this means that we need to reverse the orientation of the two open components of R, then we also reverse the orientation of all other components of R during the mutation; otherwise we do not change any orientation. For an alternative, but equivalent definition, see definition 3.3 and remark 3.4.Theorem 0.2 along with the glueing formula for ∇ s T gives rise to the following result. Corollary 0.4 (3.5) The multivariate Alexander polynomial is invariant under Conway mutation after identifying the variables corresponding to the two open strands of the mutating tangle.This result has long been known for the univariate Alexander polynomial, see for example [LM87, proposition 11], but I have been unable to find a corresponding result for the multivariate polynomial in the literature. The fact that mutation invariance follows so easily from the symmetry relations of theorem 0.2 suggests that ∇ T is well-suited for studying the "local behaviour" of the Alexander polynomial. The homological tangle invariant HFTHeegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01]. With an oriented, closed 3-dimensional manifold M , they associated a family of homological invariants, the simplest of which is denoted by HF(M). Given an oriented (null-homologous) knot or link L in M , Ozsváth and Szabó, and independently J. Rasmussen, then defined filtrations on the chain complexes which give rise to the respective flavours of knot and link Floer homology [OS03a, Ras03, OS05], the simplest of which is denoted by HFL(L). The Alexander polynomial can be recovered from these groups as the graded Euler characteristic.Given corollary 0.4, it is only natural to ask for a Heegaard-Floer theoretic categorification of ∇ s T . To this end, we define a homology theory HFT as follows: given a Heegaard diagram for a tangle T (see definition 4.1) along with a site s of T , we define a finitely generated Abelian group which comes with two gradings: a relative homological Z-grading and an Alexander grading, which is an additional relative Z-grading for each component of the tangle:CFT(T, s) = h∈Z ←homological grading a∈Z |T| ←Alexander grading CFT h (T, s, a).

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Involutive bordered Floer homology

Hendricks

Lipshitz

2019

Trans. Amer. Math. Soc.

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We give a bordered extension of involutive HF and use it to give an algorithm to compute involutive HF for general 3-manifolds. We also explain how the mapping class group action on HF can be computed using bordered Floer homology. As applications, we prove that involutive HF satisfies a surgery exact triangle and compute HFI (Σ(K)) for all 10-crossing knots K.Two years later, Manolescu and the first author introduced a shadow of Pin (2)-equivariant Seiberg-Witten Floer homology, called involutive Heegaard Floer homology [HM17], in Ozsvath-Szabó's Heegaard Floer homology [OSz04b]. Again, involutive Heegaard Floer homology has had a number of applications, again mainly to the homology cobordism group [HMZ17, BH16, DM17, Zem16].As described below, a key step in the definition of involutive Heegaard Floer homology is naturality of the Heegaard Floer invariants [OSz06, JT12]. Another implication of naturality is that the mapping class group of a 3-manifold Y acts on the Heegaard Floer invariants of Y ; this action has been studied relatively little.Bordered Heegaard Floer homology, introduced by Ozsváth, Thurston, and the second author, is an extension of the Heegaard Floer invariant HF (Y ) to 3-manifolds with boundary [LOT08, LOT15]. In particular, bordered Floer homology leads to a practical algorithm for computing HF (Y ) [LOT14b]. In this paper we extend that algorithm to compute both the hat variant of involutive Heegaard Floer homology and the mapping class group action on HF (Y ). Although the two actions are different, their description in terms of bordered Floer homology is quite similar. We also prove a (hitherto unknown) surgery exact triangle for the hat variant of involutive Heegaard Floer homology. In

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Bordered Sutured Floer Homology

Cited by 5 publications

References 19 publications

Peculiar modules for 4‐ended tangles

Peculiar modules for 4‐ended tangles

Kauffman states and Heegaard diagrams for tangles

Involutive bordered Floer homology

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