arXiv subjects
Publications and source records attributed to Artem Kotelskiy.
We show that reduced Khovanov homology over any field is invariant under component-preserving Conway mutation. Our proof relies on strong geography restrictions for a certain Khovanov multicurve invariant associated with Conway tangles that we introduced in previous work [arXiv:1910.14584]. Applying ideas from homological mirror symmetry, we give a full classification of the components of this invariant.
The earring tangle consists of four strands 4pt x [0,1] inside S^2 x [0,1] and one meridian around one of the strands. Equipping this tangle with a nontrivial SO(3) bundle, we show that its traceless SU(2) flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling -- a subtle degeneration phenomenon predicted by Bottman and Wehrheim -- appears in the context of traceless character varieties.
When $\mathbf{k}$ is a field, type D structures over the algebra $\mathbf{k}[u,v]/(uv)$ are equivalent to immersed curves decorated with local systems in the twice-punctured disk. Consequently, knot Floer homology, as a type D structure over $\mathbf{k}[u,v]/(uv)$, can be viewed as a set of immersed curves. With this observation as a starting point, given a knot $K$ in $S^3$, we realize the immersed curve invariant $\widehat{\mathit{HF}}(S^3 \smallsetminus \mathringν(K))$ [arXiv:1604.03466] by converting the twice-punctured disk to a once-punctured torus via a handle attachment. This recovers a result of Lipshitz, Ozsváth, and Thurston [arXiv:0810.0687] calculating the bordered invariant of $S^3 \smallsetminus \mathringν(K)$ in terms of the knot Floer homology of $K$.
When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal $δ$-grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant $\operatorname{HFT}$ and the Khovanov invariant $\operatorname{\widetilde{Kh}}$ that were developed by the authors in previous works.
In previous work we introduced a Khovanov multicurve invariant $\operatorname{\widetilde{Kh}}$ associated with Conway tangles. Applying ideas from homological mirror symmetry we show that $\operatorname{\widetilde{Kh}}$ is subject to strong geography restrictions: Every component of the invariant is linear, in the sense that it admits a lift to a curve homotopic to a straight line in an appropriate planar cover of the tangle boundary.
We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$. We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that $\widetilde{\operatorname{Kh}}$ and $\widetilde{\operatorname{BN}}$ detect if a Conway tangle is split.
There is a one-to-one correspondence between strong inversions on knots in the three-sphere and a special class of four-ended tangles. We compute the reduced Khovanov homology of such tangles for all strong inversions on knots with up to 9 crossings, and discuss these computations in the context of earlier work by the second author. In particular, we provide a counterexample to [Conjecture 29, arXiv:1311.1085] as well as a refinement of and additional evidence for [Conjecture 28, arXiv:1311.1085].
We compare two different types of mapping class invariants: the Hochschild homology of an $A_\infty$ bimodule coming from bordered Heegaard Floer homology, and fixed point Floer cohomology. We first compute the bimodule invariants and their Hochschild homology in the genus two case. We then compare the resulting computations to fixed point Floer cohomology, and make a conjecture that the two invariants are isomorphic. We also discuss a construction of a map potentially giving the isomorphism. It comes as an open-closed map in the context of a surface being viewed as a $0$-dimensional Lefschetz fibration over the complex plane.
Given a pointed 4-ended tangle $T \subset D^3$, there are two Khovanov theoretic tangle invariants, $\unicode{1044}_1(T)$ from [arXiv:1910.1458] and $L_T$ from [arXiv:1808.06957], which are twisted complexes over the Fukaya category of the boundary 4-punctured sphere $(S^2,4\text{pt})=\partial (D^3, T)$. We prove that these two invariants are the same.
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\operatorname{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\operatorname{BN}}(T)$ to define two immersed curve invariants $\widetilde{\operatorname{Kh}}(T)$ and $\operatorname{Kh}(T)$, which satisfy similar gluing theorems that recover reduced and unreduced Khovanov homology of links, respectively. As a first application, we prove that Conway mutation preserves reduced Bar-Natan homology over the field with two elements and Rasmussen's $s$-invariant over any field. As a second application, we give a geometric interpretation of Rozansky's categorification of the two-stranded Jones-Wenzl projector. This allows us to define a module structure on reduced Bar-Natan and Khovanov homologies of infinitely twisted knots, generalizing a result by Benheddi.
We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphic to a suitable algebraic pairing of modules M(L) and M(L'). This extends the pillowcase homology construction - given a 2-stranded tangle inside a 3-ball, if one obtains an immersed unobstructed Lagrangian inside the pillowcase, one can further associate an A infinity module to that Lagrangian.
In this paper we investigate a family of Hamiltonian-minimal Lagrangian submanifolds in ${\mathbb C}^m$, ${\mathbb C}P^m$ and other symplectic toric manifolds constructed from intersections of real quadrics. In particular, we explain the nature of this phenomenon by proving H-minimality in a more conceptual way, and prove minimality of the same submanifolds in the corresponding moment-angle manifolds.