arXiv subjects

Zhechi Cheng

Publications and source records attributed to Zhechi Cheng.

3 recordsLinked to original sources

arXiv2025

Knot Floer homology of positive braids

We compute the next-to-top term of the knot Floer homology of any link obtained as the closure of a positive braid, showing in particular that the rank is one for any prime knot in this family. As such knots are fibered, it follows that their monodromies are fixed-point free. We compare the set of positive braids with other classes of knots known to have this property. One such class consists of knots possessing "diagonal" grid diagrams. We provide an example of such a knot that is not a positive braid, providing an answer to a question of Vance and Kubota. We conclude with a number of problems and questions for future study naturally motivated by our theorem.

math.GT↗

arXiv2022

Murasugi sum and extremal knot Floer homology

The aim of this paper is to study the behavior of knot Floer homology under Murasugi sum. We establish a graded version of Ni's isomorphism between the extremal knot Floer homology of Murasugi sum of two links and the tensor product of the extremal knot Floer homology groups of the two summands. We further prove that $τ=g$ for each summand if and only if $τ=g$ holds for the Murasugi sum (with $τ$ and $g$ defined appropriately for multi-component links). Some applications are presented.

math.GT↗

arXiv2021

Bigrading the symplectic Khovanov cohomology

We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We show that it recovers the Jones grading of Khovanov homology up to an overall grading shift over any characteristic zero field, through proving that the isomorphism of Abouzaid-Smith can be refined as an isomorphism between bigraded cohomology theories. We prove it by constructing an exact triangle of symplectic Khovanov cohomology that behaves similarly to the unoriented skein exact triangle for Khovanov homology. We use a version of symplectic Khovanov cohomology defined for bridge diagrams and obtain an absolute homological grading in this construction.

math.SG↗