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Zhiyun Cheng

Publications and source records attributed to Zhiyun Cheng.

At least 19 recordsLinked to original sources
arXiv2026

GL-racks and coloring invariants of Legendrian knots

In this paper, we explore the algebraic structure of GL-racks, and demonstrate that finite GL-racks decompose canonically into permutation GL-racks and block GL-racks. As a corollary, we verify that two Legendrian knots with the same classical invariants share equivalent coloring invariants with respect to any given finite GL-rack.

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arXiv2025

When is arc crossing change an unknotting operation?

This paper extends the study of arc crossing change, a local operation on knot diagrams recently introduced by Cericola, from knot diagrams to link diagrams. We consider two types of arc crossing change on link diagrams and discuss when they are unknotting operations. Furthermore, we show that any two crossing points in an alternating knot diagram are arc crossing change admissible.

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arXiv2025

Fundamental generalized Legendrian rack and classical invariants

In this paper, we prove that if two Legendrian knots have isomorphic fundamental GL-racks, then either they have the same Thurston-Bennequin number and the same rotation number, or they have the opposite Thurston-Bennequin numbers and opposite rotation numbers.

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arXiv2025

Partial-dual genus polynomial of graphs

Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ is an enumeration of the partial duals of $G$ by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov.

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arXiv2024

The necessity of (co)unit in nearly Frobenius algebra

In this article, we concern the concept of nearly Frobenius algebra, which corresponds to most 2D-TQFT of which each cobordism admits no critical points of index 0 or 2. We prove that any nearly Frobenius algebra over a principal ideal domain with surjective multiplication and injective comultiplication is indeed a Frobenius algebra. The motivation of this study mainly emanates from the investigation of potential constructions of link homology.

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arXiv2024

A categorification for the partial-dual genus polynomial

The partial-dual genus polynomial $^\partial\varepsilon_G(z)$ of a ribbon graph $G$ is the generating function that enumerates all partial duals of $G$. In this paper, we give a categorification for this polynomial. The key ingredient of the construction is an extended Frobenius algebra related to unoriented topological quantum field theory.

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arXiv2021

Chord index for knots in thickened surfaces

In this note, we construct a chord index homomorphism from a subgroup of $H_1(Σ, \mathbb{Z})$ to the group of chord indices of a knot $K$ in $Σ\times I$. Some knot invariants derived from this homomorphism are discussed.

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arXiv2021

A categorification for the signed chromatic polynomial

By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which corresponds to signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.

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arXiv2019

Region crossing change on surfaces

Region crossing change is a local operation on link diagrams. The behavior of region crossing change on $S^2$ is well understood. In this paper, we study the behavior of (modified) region crossing change on higher genus surfaces.

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arXiv2018

Some remarks on the chord index

In this paper we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots.

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arXiv2018

A note on the writhe polynomial and the virtual crossing number

In this note we give a new lower bound on the virtual crossing number via the writhe polynomial, which refines a result of B. Mellor. The proof is based on a new interpretation of the writhe polynomial. The characterization of the writhe polynomial is also discussed.

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arXiv2018

On the Classification of Topological Quandles

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval $[0, 1]$, we conjecture that there exists only one topological quandle structure on it, i.e. the trivial one. Some evidences are provided to support our conjecture.

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