arXiv subjects

Minyi Liang

Publications and source records attributed to Minyi Liang.

3 recordsLinked to original sources

arXiv2024

Ribbon numbers of 12-crossing knots

The ribbon number of a knot is the minimum number of ribbon singularities among all ribbon disks bounded by that knot. In this paper, we build on the systematic treatment of this knot invariant initiated in recent work of Friedl, Misev, and Zupan. We show that the set of Alexander polynomials of knots with ribbon number at most four contains 56 polynomials, and we use this set to compute the ribbon numbers for many 12-crossing knots. We also study higher-genus ribbon numbers of knots, presenting some examples that exhibit interesting behavior and establishing that the success of the Alexander polynomial at controlling genus-0 ribbon numbers does not extend to higher genera.

math.GT↗

arXiv2024

Kauffman bracket skein module of two families of Seifert manifolds

We compute the Kauffman bracket skein modules of Seifert manifolds $\Sigma_{0,1}((k_1,1),(k_2,1))$ and $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ by providing presentations of them. From the obtained presentations, we show that the Kauffman bracket skein modules of $\Sigma_{0,1}((k_1,1),(k_2,1))$ are free with infinitely many generators when $k_1,k_2\ge 1$ and that of $\Sigma_{0,0}((k_1,1),(k_2,1),(k_3,1))$ are finitely generated when $k_1,k_2,k_3 \ge 2$. We also show that the empty link in either case is not trivial.

math.GT↗

arXiv2021

Set-theoretic Yang-Baxter cohomology of cyclic biquandles

We completely determine the free parts of the set-theoretic Yang-Baxter (co)homology groups of finite cyclic biquandles, along with fully computing the torsion subgroups of their 1st and 2nd homology groups. Furthermore, we provide upper bounds for the orders of torsions in the 3rd and higher dimensional homology groups. This work partially solves the conjecture that the normalized set-theoretic Yang-Baxter homology of cyclic biquandles satisfy $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}} \oplus \mathbb{Z}_{m}$ when $n$ is odd and $H_{n}^{NYB}(C_{m}) = \mathbb{Z}^{(m-1)^{n-1}}$ when $n$ is even. In addition, we obtain cocycle representatives of a basis for the rational cohomology group of a cyclic biquandle and introduce several non-trivial torsion homology classes.

math.GT↗