Recently, Kashaev and the first author constructed an R -matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin–Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence V n of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links–Gould polynomial. In this note we present the results of the computation of the V n -polynomials for n = 1 , 2 , 3 , 4 . This leads to the discovery of emerging patterns, including the genus bound for V 2 being an equality for all 352.2 million knots with at most 19 crossings, as well as unexpected Conway mutations that seem undetected by the V n -polynomials as well as by Heegaard Floer Homology and Khovanov Homology.

Let L be a null homologous link in \mathbb{RP}^{3} . We define Khovanov-type homologies of L which depend on an extra input \alpha = (V_{0},V_{1},f,g) consisting of two graded vectors spaces and two maps between them. With some specific choice of \alpha=\alpha_{\mathrm{APS}} , we recover the categorification of the Kauffman bracket due to Asaeda–Przytycki–Sikora. With another choice of \alpha = \alpha_{\mathrm{HF}} , we construct a spectral sequence from our theory converging to the Heegaard–Floer homology of the even branched double cover of \mathbb{RP}^{3} .

This article provides a review of the gauge-theoretic approach to Khovanov homology, framed in terms of a generalisation of Witten's original proposal. Concretely, the physical arguments underlying Witten's insights suggest that there is a one-parameter family of Haydys-Witten instanton Floer homology groups $HF_{\theta}\bigl(W^4\bigr)$ for four-manifolds. At the heart of the proposal is a systematic investigation of the dimensional reductions of the Haydys-Witten equations. It is shown that on the five-dimensional cylinder $M^5=\mathbb{R}_s\times W^4$ with nowhere-vanishing vector field $v=\cos\theta \partial_s+\sin\theta w$, the Haydys-Witten equations provide flow equations for the $\theta$-Kapustin-Witten equations on $W^4$. Similar reductions to lower dimensions include the twisted extended Bogomolny equations on three-manifolds and the twisted octonionic Nahm equations on one-manifolds, whose solutions provide natural boundary conditions along the boundary and corners of $W^4$. These reductions determine the indicial roots of the Haydys-Witten and $\theta$-Kapustin-Witten equations with twisted Nahm-pole boundary conditions, which are required to establish elliptic regularity. Motivated by these insights, the groups $HF_{\theta}\bigl(W^4\bigr)$ are defined in analogy with Yang-Mills instanton Floer theory: solutions of the $\theta$-Kapustin-Witten equations on $W^4$ modulo Haydys-Witten instantons on the cylinder $\mathbb{R}_s\times W^4$ interpolating between them. The relation to knot invariants observed by Witten arises when the four-manifold is the geometric blow-up $W^4=\bigl[X^3\times\mathbb{R}^+,K\bigr]$ along a knot $K\subset X^3\times{0}$ in its three-dimensional boundary. This yields a precise restatement of Witten's conjecture as the equality between $HF^\bullet_{\pi/2}\bigl(\bigl[S^3\times\mathbb{R}^+,K\bigr]\bigr)$ and Khovanov homology $\mathrm{Kh}^\bullet(K)$.

This is part one of constructing a more computationally effective spectral sequence from Khovanov homology to the δ-graded knot Floer homology. In this paper, we introduce a chain complex C1±1(K) for any link in S3 by considering a plat braid diagram for K. This complex is inspired by knot Floer homology, but the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d0+d1. We show that the E2 page of the associated spectral sequence is isomorphic to the Khovanov homology of K and that the total homology is a link invariant with a local refinement for tangles. More precisely, our complex can be refined to an invariant for braids on 2n strands, where the associated invariant is a bimodule over an algebra An. We show that An is isomorphic to B‾′(2n+1,n), the algebra used for the DA-bimodule constructed by Ozsváth and Szabó in their algebraic construction of knot Floer homology [OS16; OS19]. In the sequel paper [AD19], we show that our braid invariant is chain homotopic to the Ozsváth–Szabó’s braid invariant and that cup and cap invariants are nicely related as well.

Abstract We consider dynamically convex star‐shaped domains in a symplectic vector space of dimension 4. For such a domain, a “Hopf orbit” is a closed characteristic in the boundary which is unknotted and has self‐linking number . We show that the minimum action among Hopf orbits exists and defines a symplectic capacity for dynamically convex star‐shaped domains. We further show that this capacity agrees with the first embedded contact homology (ECH) capacity for such domains. Combined with a result of Edtmair, this implies that for dynamically convex star‐shaped domains in four dimensions, the first ECH capacity agrees with the cylinder capacity. This also provides a method to show that the first ECH capacity of a dynamically convex star‐shaped domain satisfies the axioms of a normalized symplectic capacity without any need for Seiberg–Witten theory.

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a G 2 -manifold X satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. It can also be considered as an analogue of a G 2 -instanton. In this paper, we see that some important observations that appear in other geometric problems are also found in the dDT case as follows. (1) A dDT connection exists if a 7-manifold has full holonomy G 2 and the G 2 -structure is “sufficiently large”. (2) The dDT equation is described as the zero of a certain multi-moment map. (3) The gradient flow equation of a Chern–Simons type functional of Karigiannis and Leung, whose critical points are dDT connections, agrees with the Spin ( 7 ) version of the dDT equation on a cylinder with respect to a certain metric on a certain space. This can be considered as an analogue of the observation in instanton Floer homology for 3-manifolds.

Abstract Inspired by the notions of local equivalence in monopole and Heegaard Floer homology, we introduce a version of local equivalence that combines odd Khovanov homology with equivariant even Khovanov homology into an algebraic package called a local even–odd (LEO) triple. We get a homomorphism from the smooth concordance group to the resulting local equivalence group of such triples. We give several versions of the ‐invariant that descend to , including one that completely determines whether the image of a knot in is trivial. We discuss computer experiments illustrating the power of these invariants in obstructing sliceness, both statistically and for some interesting knots studied by Manolescu–Piccirillo. Along the way, we explore several variants of this local equivalence group, including one that is totally ordered.

Recovering Generalized Homology from Floer Homology: The Complex Oriented Case

Ribbon homology cobordisms and link Floer homology

Algebras with matchings and link Floer homology

We continue our study of the integer-valued knot invariants ν ♯ ( K ) \nu ^\sharp (K) and r 0 ( K ) r_0(K) , which together determine the dimensions of the framed instanton homologies of all nonzero Dehn surgeries on K K . We first establish a “conjugation” symmetry for the decomposition of cobordism maps constructed in our earlier work, and use this to prove, among many other things, that ν ♯ ( K ) \nu ^\sharp (K) is always either zero or odd. We then apply these technical results to study linear independence in the homology cobordism group, to define an instanton Floer analogue ϵ ♯ ( K ) \epsilon ^\sharp (K) of Hom’s ϵ \epsilon -invariant in Heegaard Floer homology, and to the problem of characterizing a given 3-manifold as Dehn surgery on a knot in S 3 S^3 .

The cabling conjecture of Gonzlez-Acua and Short states that only cable knots admit Dehn surgery to a manifold containing an essential sphere.We approach this conjecture for thin knots using Heegaard Floer homology, primarily via immersed curves techniques inspired by Hanselman's work on the cosmetic surgery conjecture.We show that almost all thin knots satisfy the cabling conjecture, with a possible exception coming from a (conjecturally nonexistent) collection of thin, hyperbolic, L-space knots.This result serves as a reproof that the cabling conjecture is satisfied by alternating knots.57K18

Product and coproduct on

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A note on knot Floer homology of satellite knots with .1;1/-patterns WEIZHE SHENWe prove that if P is a .1;1/-pattern, then the two inequalities dim b HFK.P

Holomorphic polygons and the bordered Heegaard Floer homology of link complements

We prove the equivalence of the sutured versions of Heegaard Floer homology, monopole Floer homology, and embedded contact homology. As applications we show that the knot versions of Heegaard Floer homology and embedded contact homology are equivalent and that product sutured 3 -manifolds are characterized by the fact that they carry an adapted Reeb vector field without periodic orbits.

Extending work of Saneblidze–Umble and others, we use diagonals for the associahedron and multiplihedron to define tensor products of 𝐴 ∞ -algebras, modules, algebra homomorphisms, and module morphisms, as well as to define a bimodule analogue of twisted complexes (type DD structures, in the language of bordered Heegaard Floer homology) and their one- and two-sided tensor products. We then give analogous definitions for 1-parameter deformations of 𝐴 ∞ -algebras; this involves another collection of complexes. These constructions are relevant to bordered Heegaard Floer homology.

In this paper, we define the hat and tilde versions of the grid homology for spatial graphs possibly with sinks or sources by extending the grid homology developed by Harvey and O’Donnol [ 4 ]. We define a cut edge for spatial graphs and show that the grid homology for a spatial graph [Formula: see text] is trivial if [Formula: see text] has a sink, source, or cut edge. As an application, we give purely combinatorial proofs of some formulas, including a Künneth formula for the knot Floer homology of connected sums in the framework of the grid homology.

Abstract For any Legendrian knot or link in , we construct an algebra that can be viewed as an extension of the Chekanov–Eliashberg differential graded algebra. The structure incorporates information from rational symplectic field theory and can be formulated combinatorially. One consequence is the construction of a Poisson bracket on commutative Legendrian contact homology, and we show that the resulting Poisson algebra is an invariant of Legendrian links under isotopy.