arXiv subjects

Zipei Zhuang

Publications and source records attributed to Zipei Zhuang.

6 recordsLinked to original sources

arXiv2022

On the homology theory for the chromatic polynomials

In \cite{10.2140/agt.2005.5.1365}, Rong and Helme-Guizon defined a categorification for the chromatic polynomial $P_G(x)$ of graphs $G$, i.e. a homology theory $H^*(G)$ whose Euler characteristic equals $P_G(x)$. In this paper, we showed that the rational homoology $H^*(G;\mathbb{Q})$ is supported in two lines, and develop an analogy of Lee's theory for Khovanov homology. In particular, we develop a new homology theory $H_{Lee}(G)$, and showed that there is a spectral sequence whose $E_2$ -term is isomorphic to $H^*(G)$ converges to $H_{Lee}(G)$.

math.GT↗

arXiv2022

Grid homology and the unknotting number

We showed that the order of torsion homology classes in the grid homology of a knot is a lower bound for the unknotting number.

math.GT↗

arXiv2022

The dimension of Kronheimer-Mrowka instanton homology group for plane trivalent graphs

We proved that the dimension of the F-vector space J#(G) for a plane trivalent graph G, defined by Kronheimer and Mrowka using their SO(3) instanton Floer homology, is equal to the number of Tait colorings of G.

math.GT↗

arXiv2022

Knot cobordism and Lee's perturbation of Khovanov homology

For a connected cobordism S between two knots K1,K2 in S3, we establish an inequality involving the number of local maxima, the genus of S, and the torsion orders of Kht(K1),Kht(K2), where Kht denotes Lee's perturbation of Khovanov homology. This shows that the torsion order gives a lower bound for the band-unlinking number.

math.GT↗

arXiv2021

Grid homology for spatial graphs and a skein exact sequence

We defined a grid homology theory for spatial graphs. We showed that the skein exact sequence of singular knots can be extended to our grid homology for spatial graphs.

math.GT↗

arXiv2021

Ihara zeta function and twisted Alexander invariants

Lin and Wang defined a model of random walks on knot diagrams and interprete the Alexnader polynomials and the colored Jones polynomials as Ihara zeta functions, i.e. zeta functions defined by counting cycles on the knot diagram. Using this explanation, they gave a more conceptual proof for the Melvin-Morton conjecture. In this paper, we give an analogous zeta function expression for the twisted Alexander invariants.

math.GT↗