arXiv subjects

Yuanan Diao

Publications and source records attributed to Yuanan Diao.

25 records · Page 2Linked to original sources

arXiv2012

Relative Tutte polynomials of tensor products of colored graphs

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollobás and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion-contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.

math.CO↗

arXiv2011

Nullification of knots and links

In this paper, we study a geometric/topological measure of knots and links called the nullification number. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, one can intuitively regard it as a way to measure how easily a knotted circular DNA can unknot itself through recombination of its DNA strands. It turns out that there are several different ways to define such a number. These definitions lead to nullification numbers that are related, but different. Our aim is to explore the mathematical properties of these nullification numbers. First, we give specific examples to show that the nullification numbers we defined are different. We provide detailed analysis of the nullification numbers for the well known 2-bridge knots and links. We also explore the relationships among the three nullification numbers, as well as their relationships with other knot invariants. Finally, we study a special class of links, namely those links whose general nullification number equals one. We show that such links exist in abundance. In fact, the number of such links with crossing number less than or equal to n grows exponentially with respect to n.

math.GT↗

arXiv2009

The Ropelengths of Knots Are Almost Linear in Terms of Their Crossing Numbers

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number.

math.GT↗

arXiv2009

Relative Tutte Polynomials for Colored Graphs and Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

math.CO↗

arXiv2009

Quadrisecants give new lower bounds for the ropelength of a knot

Using the existence of a special quadrisecant line, we show the ropelength of any nontrivial knot is at least 15.66. This improves the previously known lower bound of 12. Numerical experiments have found a trefoil with ropelength less than 16.372, so our new bounds are quite sharp.

math.GT↗

arXiv2005

Weyl-Heisenberg Frame Wavelets with Basic Supports

Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset \mathbb{R}$ is called a Weyl-Heisenberg frame set for $(a, b)$ if the function $g=χ_{E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. It is an open question on how to characterize all frame sets for a given pair $(a,b)$ in general. In the case that $a=2π$ and $b=1$, a result due to Casazza and Kalton shows that the condition that the set $F=\bigcup_{j=1}^{k}([0,2π)+2n_{j}π)$ (where $\{n_{1}<n_{2}<...<n_{k}\}$ are integers) is a Weyl-Heisenberg frame set for $(2π,1)$ is equivalent to the condition that the polynomial $f(z)=\sum_{j=1}^{k}z^{n_{j}}$ does not have any unit roots in the complex plane. In this paper, we show that this result can be generalized to a class of more general measurable sets (called basic support sets) and to set theoretical functions and continuous functions defined on such sets.

math.FA↗

arXiv2005

From Weyl-Heisenberg Frames to Infinite Quadratic Forms

Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special forms and certain strongly positive definite quadratic forms of infinite dimension. Some examples of application in matrix algebra are provided.

math.FA↗