arXiv research

Study calculates special knot properties for specific types of knots.

Formula derived for cabled knots' concordance invariants.

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

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and …

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

Topological free involutions on S^1xS^n are classified up to conjugation. As a byproduct we obtain a new computation of the group of concordance classes of homeomorphisms of the projective space RP^n.

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.

New invariant detects infinite order cabled knots.

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string reversal, then T/Fix(R) contains an infinitely generated free subgroup. The resu…

New knots not rationally concordant to their reverses found.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

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

New method improves HMC for sampling on manifolds.

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 $\bar{d}$…

Study on equivariant Q-sliceness for strongly invertible knots.

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a $\mathbb{Z}^\infty$-subgroup and a $\mathbb{Z}$-summand. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of He…

The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.

Classifies involutions on spherical 3-manifolds.

Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.

Stability theorem for concordance embeddings with applications.

We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Yx[0,1] and the action of the concordance group of knots in the 3-sphere that ties in local knots. We prove that the trivial fre…

The paper studies circular evolutes and involutes of framed curves in Euclidean space.

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11--crossing prime knots. This analysis resolves the concordance ge…

This work shows non-abelian quotient groups in string link concordance.

New formula for dual knots using involutions.

The study classifies involutions on del Pezzo surfaces.

Characterizes the Legendre involution on generic frontals.

Khovanov homology shows (4,5) torus knot is a summand in its concordance class.

Study shows knots with similar Blanchfield forms can be homotopy ribbon concordant.

The paper defines conditions for good involutions in generalized Alexander quandles.

New homomorphism from Khovanov homology for knot concordance.

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

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…

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

The study proves symplectic quandles cannot have good involutions.

Kirby and Lickorish showed that every knot in the 3-sphere is concordant to a prime knot, equivalently, every concordance class contains a prime knot. We prove here that their result can be strengthened: Every knot in the 3-sphere is invertibly concordant to a prime knot. A consequence is that every double concordance …

Abstract: Proves non-abelian group of equivariant concordance.

The concordance genus of a knot is the least genus of any knot in its concordance class. Although difficult to compute, it is a useful invariant that highlights the distinction between the three-genus and four-genus. In this paper we define and discuss the stable concordance genus of a knot, which describes the behavio…

Introduces slice knots and concordance, linking to exotic smooth structures.

We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a…

Involutions generate mapping class groups of infinite surfaces.

New hyperbolic knots not concordant to algebraic ones found.

Minimal involutions generate a subgroup of nonorientable surfaces.

Study exact surgery formula in involutive Heegaard Floer homology.

We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manif…

We define an infinite family of linearly independent, integer-valued smooth concordance homomorphisms. Our homomorphisms are explicitly computable and rely on local equivalence classes of knot Floer complexes over the ring $\mathbb{F}[U, V]/(UV=0)$. We compare our invariants to other concordance homomorphisms coming fr…

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.

A fibered concordance of knots, introduced by Harer, is a concordance between fibered knots that is well-behaved with respect to the fibrations. We consider semi-fibered concordance of two component ordered links $L=J \sqcup K$ with $J$ fibered. These are concordances that restrict to fibered concordances on the first …