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Wenzhao Chen

Publications and source records attributed to Wenzhao Chen.

12 recordsLinked to original sources
arXiv2025

Involutions on S^4

This paper studies locally linear involutions on S^4. Our main theorem shows that any such involution with a 1-dimensional fixed-point set is necessarily linear, provided the fixed-point set admits an equivariant tubular neighborhood. The proof combines modified surgery theory with an equivariant version of the Schoenflies theorem, which we establish here. We also show that equivariant tubular neighborhoods of 1-dimensional fixed-point sets, when they exist, are not unique, in contrast to the nonequivariant case. Our results combine with earlier work to provide a classification of all locally linear involutions on S^4. As a further application, we obtain that strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice with respect to the linear action, strengthening a previous result of the first two authors. Finally, we also prove that when the fixed-point set is 2-dimensional, the involution is linear if and only if the fixed-point set is an unknotted 2-knot.

math.GT↗
arXiv2024

Equivariant unknotting numbers of strongly invertible knots

We study symmetric crossing change operations for strongly invertible knots. Our main theorem is that the most natural notion of equivariant unknotting number is not additive under connected sum, in contrast with the longstanding conjecture that unknotting number is additive.

math.GT↗
arXiv2023

Negative amphichiral knots and the half-Conway polynomial

In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize the factor $f(z)$ to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in $S^3$, answering a problem of Hartley-Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group.

math.GT↗
arXiv2023

Satellite knots and immersed Heegaard Floer homology

We describe a new method for computing the $UV = 0$ knot Floer complex of a satellite knot given the $UV = 0$ knot Floer complex for the companion and a doubly pointed bordered Heegaard diagram for the pattern, showing that the complex for the satellite can be computed from an immersed doubly pointed Heegaard diagram obtained from the Heegaard diagram for the pattern by overlaying the immersed curve representing the complex for the companion. This method streamlines the usual bordered Floer method of tensoring with a bimodule associated to the pattern by giving an immersed curve interpretation of that pairing, and computing the module from the immersed diagram is often easier than computing the relevant bordered bimodule. In particular, for (1,1) patterns the resulting immersed diagram is genus one, and thus the computation is combinatorial. For (1,1) patterns this generalizes previous work of the first author which showed that such immersed Heegaard diagram computes the $V=0$ knot Floer complex of the satellite. As a key technical step, which is of independent interest, we extend the construction of a bigraded complex from a doubly pointed Heegaard diagram and of an extended type D structure from a torus-boundary bordered Heegaard diagram to allow Heegaard diagrams containing an immersed alpha curve.

math.GT↗
arXiv2020

An infinite-rank summand from iterated Mazur pattern satellite knots

We show there exists a topologically slice knot $K$ such that the knots $\{M^n(K)\}_{n=0}^\infty$ obtained by iterated satellite operations by the Mazur pattern span an infinite-rank summand of the smooth knot concordance group. This answers a question raised by Feller-Park-Ray.

math.GT↗
arXiv2020

A lower bound for the double slice genus

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category.

math.GT↗
arXiv2019

Knot Floer homology of satellite knots with (1,1)-patterns

For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the $τ$-invariant. For patterns $P$ obtained from two-bridge links $b(p,q)$, we derive a formula for the $τ$-invariant of $P(T_{2,3})$ and $P(-T_{2,3})$ in terms of $(p,q)$, and use this formula to study whether such patterns induce homomorphisms on the concordance group, providing a glimpse at a conjecture due to Hedden.

math.GT↗
arXiv2018

On the Alexander polynomial and the signature invariant of two-bridge knots

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa-Murasugi conjecture for two-bridge knots.

math.GT↗
arXiv2017

A note on 0-bipolar knots of concordance order two

Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by Hedden, Kim, and Livingston to prove $\mathbb{Z}_2^{\infty} < \mathcal{T}$ can be used to see $\mathbb{Z}_2^{\infty} < \mathcal{T}_0/\mathcal{T}_1$.

math.GT↗
arXiv2016

On the Upsilon invariant of cable knots

In this paper, we study the behavior of $Υ_K(t)$ under the cabling operation, where $Υ_K(t)$ is the knot concordance invariant defined by Ozsváth, Stipsicz, and Szabó, associated to a knot $K\subset S^3$. The main result is an inequality relating $Υ_K(t)$ and $Υ_{K_{p,q}}(t)$, which generalizes the inequalities of Hedden and Van Cott on the Ozsváth-Szabó $τ$-invariant. As applications, we give a computation of $Υ_{(T_{2,-3})_{2,2n+1}}(t)$ for $n\geq 8$, and we also show that the set of iterated $(p,1)$-cables of $Wh^{+}(T_{2,3})$ for any $p\geq 2$ span an infinite-rank summand of topologically slice knots.

math.GT↗
arXiv2012

Mod r Vanishing Theorem of Seiberg-Witten Invariant for 4-Manifolds acted by Cyclic Group Z_r

In this paper, a vanishing theorem is stated and proved. If a 4-manifold $M$ admits a smooth action by a cyclic group $\mathbb{Z}_r$, then given an $\mathbb{Z}_r$-equivariant $Spin^c$-structure $\mathcal{C}$ on $M$, the Seiberg-Witten invariant $SW\mathcal{(C)}$ is zero modulo $r$ under some slight assumptions. Here $r$ can be any positive integer. This theorem is a generalization of the mod p vanishing theorem when $\mathbb{Z}_p$ is prime order cyclic proved by F.Fang.

math.DG↗