arXiv subjects

Weizhe Shen

Publications and source records attributed to Weizhe Shen.

3 recordsLinked to original sources

arXiv2023

Persistent Legendrian contact homology in $\mathbb{R}^3$

This work applies the ideas of persistent homology to the problem of distinguishing Legendrian knots. We develop a persistent version of Legendrian contact homology by filtering the Chekanov-Eliashberg DGA using the action (height) functional. We present an algorithm for assigning heights to a Lagrangian diagram of a Legendrian knot, and we explain how each Legendrian Reidemeister move changes the height of generators of the DGA in a way that is predictable on the level of homology. More precisely, a Reidemeister move that changes an area patch of a Lagrangian diagram by δ will induce a 2δ-interleaving on the persistent Legendrian contact homology, computed before and after the Reidemeister move. Finally, we develop strong Morse inequalities for our persistent Legendrian contact homology.

math.SG↗

arXiv2023

On $χ-$slice pretzel links

A link is called $χ-$slice if it bounds a smooth properly embedded surface in the 4-ball with no closed components and Euler characteristic 1. If a link has a single component, then it is $χ-$slice if and only if it is slice. One motivation for studying such links is that the double cover of the 3-sphere branched along a nonzero determinant $χ-$slice link is a rational homology 3-sphere that bounds a rational homology 4-ball. This article aims to generalize known results about the sliceness of pretzel knots to the $χ-$sliceness of pretzel links. In particular, we completely classify positive and negative pretzel links that are $χ-$slice, and obtain partial classifications of 3-stranded and 4-stranded pretzel links that are $χ-$slice. As a consequence, we obtain infinite families of Seifert fiber spaces that bound rational homology 4-balls.

math.GT↗

arXiv2022

A note on knot Floer homology of satellite knots with (1,1)-patterns

We prove that if $P$ is a $(1,1)$-pattern knot, the two inequalities $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (P(U))$ and $\dim \widehat{HFK} (P(K)) \geqslant \dim \widehat{HFK} (K)$ hold for the unknot $U\subset S^3$ and any companion knot $K\subset S^3$.

math.GT↗