arXiv subjects

Shunyu Wan

Publications and source records attributed to Shunyu Wan.

10 recordsLinked to original sources

arXiv2026

Heegaard Floer knot trace invariants, exotic 4-manifolds, and symplectic obstructions

We show that the numerical invariants $\nu$ and $|\varepsilon|$ coming from knot Floer homology are knot $n$-trace invariants for any integer $n$, resolving the remaining case in Hayden-Mark-Piccirillo. This extension allows us to construct new families of exotic pairs using Yasui patterns. Moreover, by studying the invariant $\widehat{\nu}(K)=|\varepsilon(K)|(2\nu(K)-1)$, we give a new topological obstruction to a $4$-manifold being a strong symplectic filling of any contact structure on the boundary.

math.GT↗

arXiv2026

The next-to-top term of the knot Floer homology of some non-fibered knots

Sivek conjectured that the rank of knot Floer homology in the next-to-top Alexander grading is at least the rank in the top Alexander grading. Baldwin and Vela-Vick verified this conjecture in the case of fibered knots arXiv:1801.06563. Ni gave a generalization of this result (for knots in generalized $L$-spaces) to cases in which the knot Floer homology satisfies an algebraic condition arXiv:2104.14687. We give an independent generalization of Baldwin and Vela-Vick's result to a family of knots with Seifert surfaces satisfying certain conditions.

math.GT↗

arXiv2026

Exotic $\mathbb{R}^4$'s, RBG Links, and End Floer Homology

We give the first pair of non-diffeomorphic exotic $\mathbb{R}^4$'s made by attaching diffeomorphic Casson handles onto diffeomorphic disk complements. Our examples are obtained using the RBG link construction to find slice knots with diffeomorphic slice disk complements, but whose Whitehead doubled disk complements are not diffeomorphic. We distinguish the exotic $\mathbb{R}^4$'s using end Floer homology.

math.GT↗

arXiv2025

Concentration structures on categories and horizontal categorification

We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.

math.CT↗

arXiv2025

Surgeries on knots and tight contact structures

For any knot $K$ in $S^3$ and any positive rational $r$, we show that smooth $(-r)$-surgery on $K$ always admits a tight contact structure. More specifically, the tightness is detected by the non-vanishing Heegaard Floer contact invariant.

math.GT↗

arXiv2024

Half grid diagrams and Thompson links

We define half grid diagrams and prove every link is half grid presentable by constructing a canonical half grid pair (which gives rise to a grid diagram of some special type) associated with an element in the oriented Thompson group. We show that this half grid construction is equivalent to Jones' construction of oriented Thompson links. Using this equivalence, we relate the (oriented) Thompson index to several classical topological link invariants, and give both the lower and upper bounds of the maximal Thurston-Bennequin number of a knot in terms of the oriented Thompson index. Moreover, we give a one-to-one correspondence between half grid diagrams and elements in symmetric groups and give a new description of link group using two elements in a symmetric group.

math.GT↗

arXiv2024

On Legendrian representatives of non-fibered knots

We show that in $(S^3,\xi_{std})$ if $K$ is a non-trivial knot that realizes the three-dimensional Thurston-Bennequin bound (i.e. $K$ has a Legendrian representative $\Lambda$ with $tb(\Lambda)-rot(\Lambda)=2g(K)-1$), then $K$ has a Legendrian representative $L$ with $tb=0$. Moreover, this result can be easily generalized to contact manifolds that uniquely represent the associated contact invariants. This is the first result on Legendrian representatives of non-fibered knots in $3-$manifolds other than $S^3$. We also show that if $K$ is a nearly fibered knot in $S^3$ then $\tau(K)=g(K)$ implies that $K$ realizes the three-dimensional Thurston-Bennequin bound.

math.GT↗

arXiv2024

Negative contact surgery on Legendrian non-simple knots

We prove that for any pair of Legendrian representatives of the Chekanov-Eliashberg twist knots with different LOSS invariants, any negative rational contact $r$-surgery with $r\neq -1$ always gives rise to different contact 3-manifolds distinguished by their contact invariants. This gives the first examples of pairs of Legendrian knots with the same classical invariants but distinct contact $r$-surgeries for all negative rational number $r$. We also generalize the statement from the twist knots to a certain families of two-bridge knots.

math.GT↗

arXiv2023

Tight contact structures on some families of small Seifert fiber spaces

Suppose $K$ is a knot in a 3-manifold $Y$, and that $Y$ admits a pair of distinct contact structures. Assume that $K$ has Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is $-\Sigma(2,3,6m+1)$ and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces.

math.GT↗

arXiv2023

Naturality of Legendrian LOSS invariant under positive contact surgery

Ozsvath and Stipsicz showed that the LOSS invariant is natural under +1 contact surgery. We extend their result and prove the naturality of the LOSS invariant of a Legendrian L under any positive integer contact surgery along another Legendrian S . In addition, when S is rationally null-homologous, we also entirely characterize the Spin^c structure in the surgery cobordism that makes the naturality of contact invariant or LOSS invariant (without conjugation ambiguity). In particular this implies that contact invariant of the +n contact surgery along a rationally null-homologous Legendrian S depends only on the classical invariants of S. The additional generalityprovided by those results allows us to prove that if two Legendrian knots have different LOSS invariants then after adding the same positive twists to each in a suitable sense, the two new Legendrian knots will also have different LOSS invariants. This leads to new infinite families of examples of Legendrian (or transverse) non-simple knots that are distinguished by their LOSS invariants.

math.GT↗