arXiv research

Study exact surgery formula in involutive Heegaard Floer homology.

Study proves naturality and functoriality in a type of Heegaard Floer homology.

Extends Khovanov homology spectral sequence using Heegaard Floer homology.

Floer homology vanishes on certain 3-manifolds formed by knots and their mirrors.

Flip symmetry on knot diagrams affects Khovanov homology.

We compute the involutive Heegaard Floer homology of the family of three-manifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involu…

Study shows how knot Floer homology and bordered Floer theory are linked.

New bounds on knot unknotting numbers using involutive homology.

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homolo…

This paper introduces a new homology theory for Yang-Baxter solutions.

The paper develops a new Floer theory for 3-manifolds with involutions.

We give a bordered extension of involutive HF-hat and use it to give an algorithm to compute involutive HF-hat for general 3-manifolds. We also explain how the mapping class group action on HF-hat can be computed using bordered Floer homology. As applications, we prove that involutive HF-hat satisfies a surgery exact t…

The paper calculates involutive Heegaard Floer homology for specific 3-manifolds.

New invariants detect corks obstructing homology ball extensions.

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen's connected Seiberg-Witten Floer homology. We use this invariant to st…

New real invariants for 3-manifolds and links.

We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.

We show that a certain involution on the well known homology sphere (the boundary of the Mazur manifold) induces a nontrivial homomorphism on its Heegard-Floer homology groups (recently defined by Ozsvath and Szabo). We discuss a possible application of this to constructing exotic smooth structures on 4-manifolds.

We study the conjugation involution in Seiberg-Witten theory in the context of the Ozsváth-Szabó and Bloom's spectral sequence for the branched double cover of a link $L$ in $S^3$. We prove that there exists a spectral sequence of $\mathbb{F}[Q]/Q^2$-modules (where $Q$ has degree $-1$) which converges to $\widetilde{\m…

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to $\mathbb{Z}_4$-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism, $\underline{d}$ and $\bar{d}$…

Develops Floer cohomology for 4-manifolds with involutions and links.

New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.

A new surgery formula for knot lattice homology.

New invariants prove exotic slice disks for knots.

Real bordered Floer homology computes 3-manifolds with involution.

New invariants improve Heegaard Floer slice genus and clasp number bounds.

Proves spectral sequence for real Heegaard Floer homology.

Study of knot Floer homology and its relation to Heegaard Floer homology via equivariant surgery.

We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a coro…

New invariants for 4-manifolds obstruct certain surface pairs.

This note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements". The mistakes are the result of the incorrect assumption that for a knot $K$ inside a homology sphere $Y$, the involution on the knot Floer homology of $K$ which corresponds to moving the basepoints by o…

We prove that if two knots are concordant, their involutive knot Floer complexes satisfy a certain type of stable equivalence.

Real Heegaard Floer Homology extends Li's real monopole Floer homology.

Study symmetries in equivariant Khovanov homology.

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some invol…

New spin on Khovanov-Rozansky homology categorifies spin link polynomial.

New invariants from Seiberg-Witten theory for 3-spheres with involution.

New homomorphism proven using immersed curves on disks.

We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…

New method connects knot Floer homology with bordered Floer homology.

Develops equivariant grid homology for strongly invertible knots.

Real Heegaard Floer homology gets a new grading for certain 3-manifolds.

Using the covering involution on the double branched cover of the three-sphere branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots.

The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homolog…

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As…

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$-equivaria…

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…