New Structures in Low-Dimensional Topology Workshop 2026

The flip symmetry on knot diagrams induces an involution on Khovanov homology, and a folklore conjecture asserts that this map is actually trivial, at least with F_2 coefficients. We confirm this conjecture by adapting techniques from Alishahi-Truong-Zhang and Rozansky-Willis. I will explain the idea of the proof as well as some corollaries on involutive Khovanov homology and involutions on the Khovanov homology of strongly invertible knots. This is joint work with Daren Chen.

A pattern knot in a solid torus defines a self-map of the smooth knot concordance group. We prove that if the winding number of a pattern is even but not divisible by 8, then the corresponding map is not a homomorphism, thus partially establishing a conjecture of Hedden. I will discuss Lidman, Miller, and Pinzon-Caicedo’s branched cover obstruction to homomorphism inducing patterns. I will then use the branched cover obstruction to prove that a pattern does not induce a homomorphism in the case of winding number $w=2\mod 4$.

The Laplacian matrix of a graph is given by the degree matrix minus the adjacency matrix. The spectrum of this matrix gives us information about the graph; its study is additionally motivated by interest in the Laplacian of Riemann surfaces. When these degenerate, i.e. close to the boundary of the modular space, their associated eigenvalues approach those of the discrete Laplacian on a graph obtained from the pair of pants decomposition of the surface. I provide a complete description of the sets of eigenvalues of the weighted graph Laplacian for all graphs on four vertices that correspond to a valid pair of pants decomposition of a surface of genus 3. I also present a result for complete graphs of any size, and discuss further ideas, including connections to graph suspensions. This is joint work with Alena Erchenko and Dmitry Jakobson.

We will review some results about Khovanov Laplacians and introduce their motivation grounded in spectral geometry and quantum algorithms. This is joint work with A. Lauda.

Invariants of knotted surfaces coming from real Seiberg-Witten gauge theory have drawn recent attention due to their apparent utility for distinguishing exotic pairs. What further information can we glean from such invariants? We will discuss recent work that takes a step in the direction of this question, and use real Seiberg-Witten theory to say something about the topology of possible Seifert solids for 2-knots in the 4-sphere.

Recently introduced by Guth and Manolescu, real Heegaard Floer homology is an invariant associated to 3-manifolds equipped with an involution. In this short talk, we’ll specialize to double branched covers of links in S^3, in which case the Euler characteristic is the Heegaard Floer analogue of Miyazawa’s degree invariant. We’ll see that it can be computed inductively using a skein relation, and that it can be expressed in terms of the Alexander polynomial of the link.

Real Heegaard Floer homology is an invariant recently defined by Gary Guth and Ciprian Manolescu, which takes as input a 3-manifold equipped with a branched cover involution. It is motivated by the desire to obtain combinatorially computable analogues of Jiakai Li's Real Monopole homology and Jin Miyazawa's Real Seiberg-Witten invariants, which were used by the latter to detect knotted projective spaces in S^4. In this talk, we discuss the issue of upgrading to Z-coefficients (which was left open by Guth-Manolescu's original paper), and its relevance to obtaining applications of similar strength to Miyazawa's results.

TQFT constructions based on non-semisimple ribbon categories are well-known to produce interesting invariants of 3-manifolds. More recently, similar strategies have been applied to 4-manifolds, notably leading to Kerler-Lyubashenko functors introduced by Beliakova-De Renzi and skein (3+1)-TQFTs due to Constantino-Geer-Haioun-Patureau-Mirand. In this talk I will explain how under appropriate assumptions invariants of 4-dimensional 2-handlebodies up to 2-equivalence deriving from these two constructions coincide.

The Khovanov-Rozansky skein lasagna module was introduced by Morrison-Walker-Wedrich as an invariant of 4-manifold with a framed oriented link in the boundary. I will discuss an extension of the skein lasagna theory to 4-manifolds with codimension 2 corners, and its behavior under gluing. I will also talk about a categorical framework for computing skein lasagna modules of closed 4-manifolds via trisection, as well as an extended 4d TQFT based on skein lasagna theory. This is joint work with Sarah Blackwell and Slava Krushkal.

We construct non-split, 2-component links of Mazur manifolds in S^4, as well as links in #^n CP^2 which are split topologically but not smoothly. As a consequence, we obtain exotic embeddings of many Mazur manifolds into S^4, as well as exotic pairs of definite 4-manifolds with boundary.