arXiv subjects

Xian'an Jin

Publications and source records attributed to Xian'an Jin.

64 records · Page 4Linked to original sources

arXiv2018

The generalized Yamada polynomials of virtual spatial graphs

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is similar to the topological definition of a virtual link. Our main goal is to generalize the classical Yamada polynomial that is defined for a spatial graph. We define a generalized Yamada polynomial for a virtual spatial graph and prove that it can be normalized to a rigid vertex isotopic invariant and to a pliable vertex isotopic invariant for graphs with maximum degree at most 3. We consider the connection and difference between the generalized Yamada polynomial and the Dubrovnik polynomial of a classical link. The generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for virtual links such that it can be used to detect the non-classicality of some virtual links. We obtain a specialization for the generalized Yamada polynomial (via the Jones-Wenzl projector $P_2$ acting on a virtual spatial graph diagram), that can be used to write a program for calculating it based on Mathematica Code.

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arXiv2018

Graphical virtual links and a polynomial of signed cyclic graphs

For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virtual link L is graphical if and only if it is checkerboard colorable. On the other hand, we introduce a polynomial F[G] for signed cyclic graphs, which is defined via a deletion-marking recursion. We shall establish the relationship between F[G] of a signed cyclic graph G and the bracket polynomial of one of the virtual link diagrams associated with G. Finally we give a spanning subgraph expansion for F[G].

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arXiv2017

The Minimal Coloring Number Of Any Non-splittable $\mathbb{Z}$-colorable Link Is Four

K. Ichihara and E. Matsudo introduced the notions of $\mathbb{Z}$-colorable links and the minimal coloring number for $\mathbb{Z}$-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable $\mathbb{Z}$-colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable $\mathbb{Z}$-colorable link is exactly 4.

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arXiv2017

Characterizations of Eulerian and even-face partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, we shall first extend Huggett and Moffatt's result to any orientable ribbon graph and provide an example to show that it is not true for non-orientable ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are much more simple than semi-crossing directions.

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arXiv2017

Several extreme coefficients of the Tutte polynomial of graphs

Let $t_{i,j}$ be the coefficient of $x^iy^j$ in the Tutte polynomial $T(G;x,y)$ of a connected bridgeless and loopless graph $G$ with order $n$ and size $m$. It is trivial that $t_{0,m-n+1}=1$ and $t_{n-1,0}=1$. In this paper, we obtain expressions of another eight extreme coefficients $t_{i,j}$'s with $(i,j)=(0,m-n)$,$(0,m-n-1)$,$(n-2,0)$,$(n-3,0)$,$(1,m-n)$,$(1,m-n-1)$,$(n-2,1)$ and $(n-3,1)$ in terms of small substructures of $G$. Among them, the former four can be obtained by using coefficients of the highest, second highest and third highest terms of chromatic or flow polynomials, and vice versa. We also discuss their duality property and their specializations to extreme coefficients of the Jones polynomial.

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arXiv2015

A formula for the number of the spanning trees of line graphs

Let $G=(V,E)$ be a loopless graph and $\mathcal{T}(G)$ be the set of all spanning trees of $G$. Let $L(G)$ be the line graph of the graph $G$ and $t(L(G))$ be the number of spanning trees of $L(G)$. Then, by using techniques from electrical networks, we obtain the following formula: $$ t(L(G)) = \frac{1}{\prod_{v\in V}d^2(v)}\sum_{T\subseteq \mathcal{T}(G)}\big[\prod_{e = xy\in T}d(x)d(y)\big]\big[\prod_{e = uv\in E\backslash T}[d(u)+d(v)]\big]. $$ As a result, we provide a very simple and different proof of the formula on the number of spanning trees of some irregular line graphs, and give a positive answer to a conjecture proposed by Yan [J. Combin. Theory Ser. A 120 (2013) no. 7, 1642-1648]. By applying our formula we also derive the number of spanning trees of circulant line graphs.

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arXiv2015

Minimal sufficient sets of colors and minimum number of colors

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.

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arXiv2014

When will the crossing number of an alternating link decrease by two via a crossing change?

Let $D$ be a reduced alternating diagram of a non-split link $L$ and $\tilde{L}$ be the link whose diagram is obtained from $D$ by a crossing change. If $\tilde{L}$ is alternating, then $c(\tilde{L})\leq c(L)-2$. In this paper we explore when $c(\tilde{L})=c(L)-2$ holds and obtain a simple sufficient and necessary condition in terms of plane graphs corresponding to $L$. This result is obtained via analyzing the behavior of the Tutte polynomial of the signed plane graph corresponding to $\tilde{L}$.

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arXiv2012

A relationship between two graphical models of the Kauffman polynomial

There are two oriented 4-valent graphical models for the Kauffman polynomial: one (HJ) is obtained by combining Jaeger's formula and Kauffman-Vogel model for the Homflypt polynomial; the other (WF) is obtained by combining Kauffman-Vogel model for the Kauffman polynomial and Wu's formula. The main goal of this paper is to explore the relationship between the two models. We find that there is an one-to-many correspondence between the terms of HJ model and the terms of WF model. In addition, we investigate the relation between trivalent graphical models and 4-valent graphical models of both the Homflypt and Kauffman polynomials, and observe that there is a bijection between the terms of the two models.

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arXiv2005

The replacements of signed graphs and Kauffman brackets of links

Let $G$ be a signed graph. Let $\hat{G}$ be the graph obtained from $G$ by replacing each edge $e$ by a chain or a sheaf. We first establish a relation between the $Q$-polynomial of $\hat{G}$[6] and the $W$-polynomial of $G$ [9]. Two special dual cases are derived from the relation, one of which has been studied in [8]. Based on the one to one correspondence between signed plane graphs and link diagrams, and the correspondence between the $Q$-polynomial of signed plane graph and the Kauffman bracket of link diagram, we can compute the Kauffman bracket of link diagram corresponding to $\hat{G}$ by means of the $W$-polynomial of $G$. By this way we use transfer matrix approach to compute the Kauffman bracket of rational links, and obtain their closed-form formulae. Finally we provide an example to point out that the relation we built can be used to deal with a wide type of link family.

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