arXiv subjects

Yuanan Diao

Publications and source records attributed to Yuanan Diao.

At least 19 recordsLinked to original sources

arXiv2026

New formulas for the Jones polynomial of a rational link

We derive new formulas for the Jones polynomial and the Kauffman bracket polynomial of a rational link represented by a standard diagram that is not necessarily alternating. These formulas generalize the results of Qazaqzeh, Yasein, and Abu-Qamar for the Tutte polynomial of the Tait graph of an alternating diagram of a rational link, as well as the matrix formulas of Lawrence and Rosenstein for the Jones polynomial of a rational link. Our approach uses the colored version of Brylawski's tensor product formula for Tutte polynomials of colored graphs, due to Diao, Hetyei, and Hinson. Furthermore, generalizing the formulas of Qazaqzeh, Yasein, and Abu-Qamar, we present a finite automaton that computes the crossing signs, thereby enabling the calculation of the writhe of a standard diagram of a rational link.

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arXiv2025

The Enumeration of Alternating Pretzel Links

In this paper, we tabulate the set of alternating pretzel links. Specifically, for any given crossing number $c$, we derive a closed formula that would allow us to compute $\mathcal{P}(c)$, the total number of alternating pretzel links with crossing number $c$. Numerical computation suggests that $\mathcal{P}(c)\approx 0.155e^{0.588c}$. That is, the number of alternating pretzel links with a given crossing number $c$ grows exponentially in terms of $c$.

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arXiv2024

The Braid Indices of Pretzel Links: A Comprehensive Study, Part II

This paper is the second part of our comprehensive study on the braid index problem of pretzel links. Our ultimate goal is to completely determine the braid indices of all pretzel links, alternating or non alternating. In our approach, we divide the pretzel links into three types as follows. Let $D$ be a standard diagram of an oriented pretzel link $\mathcal{L}$, $S(D)$ be the Seifert circle decomposition of $D$, and $C_1$, $C_2$ be the Seifert circles in $S(D)$ containing the top and bottom long strands of $D$ respectively, then $\mathcal{L}$ is classified as a Type 1 (Type 2) pretzel link if $C_1\not=C_2$ and $C_1$, $C_2$ have different (identical) orientations. In the case that $C_1=C_2$, then $\mathcal{L}$ is classified as a Type 3 pretzel link. In our previous paper, we succeeded in reaching our goal for all Type 1 and Type 2 pretzel links. That is, we successfully derived precise braid index formulas for all Type 1 and Type 2 pretzel links. In this paper, we present the results of our study on Type 3 pretzel links. In this case, we are very close to reaching our goal. More precisely, with the exception of a small percentage of Type 3 pretzel links, we are able to determine the precise braid indices for the majority of Type 3 pretzel links. Even for those exceptional ones, we are able to determine their braid indices within two consecutive integers. With some numerical evidence, we conjecture that in such a case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas.

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arXiv2024

The Braid Indices of Pretzel Links: A Comprehensive Study, Part I

The determination of the braid index of an oriented link is generally a hard problem. In the case of alternating links, some significant progresses have been made in recent years which made explicit and precise braid index computations possible for links from various families of alternating links, including the family of all alternating Montesinos links. However, much less is known for non-alternating links. For example, even for the non-alternating pretzel links, which are special (and simpler) Montesinos links, the braid index is only known for a very limited few special cases. In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. We classify the pretzel links into three different types according to the Seifert circle decompositions of their standard link diagrams. More specifically, if $D$ is a standard diagram of an oriented pretzel link $\mathcal{L}$, $S(D)$ is the Seifert circle decomposition of $D$, and $C_1$, $C_2$ are the Seifert circles in $S(D)$ containing the top and bottom long strands of $D$ respectively, then $\mathcal{L}$ is classified as a Type 1 (Type 2) pretzel link if $C_1\not=C_2$ and $C_1$, $C_2$ have different (identical) orientations. In the case that $C_1=C_2$, then $\mathcal{L}$ is classified as a Type 3 pretzel link. In this paper, we present the results of our study on Type 1 and Type 2 pretzel links. Our results allow us to determine the precise braid index for any non-alternating Type 1 or Type 2 pretzel link. Since the braid indices are already known for all alternating pretzel links from our previous work, it means that we have now completely determined the braid indices for all Type 1 and Type 2 pretzel links.

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arXiv2023

The Braid Indices of the Reverse Parallel Links of Alternating Knots

The braid indices of most links remain unknown as there is no known universal method that can be used to determine the braid index of an arbitrary knot. This is also the case for alternating knots. In this paper, we show that if $K$ is an alternating knot, then the braid index of any reverse parallel link of $K$ can be precisely determined. More precisely, if $D$ is a reduced diagram of $K$, $v_+(D)$ ($v_-(D)$) is the number of regions in the checkerboard shading of $D$ for which all crossings are positive (negative), $w(D)$ is the writhe of $D$, then the braid index of a reverse parallel link of $K$ with framing $f$, denoted by $\mathbb{K}_f$, is given by the following precise formula $$\textbf{b}(\mathbb{K}_f)=\left\{ \begin{array}{ll} c(D)+2+a(D)-f, &\ {\rm if}\ f < a(D),\\ c(D)+2, &\ {\rm if}\ a(D)\le f \le b(D),\\ c(D)+2-b(D)+f, &\ {\rm if}\ f > b(D),\\ \end{array} \right. $$ where $a(D)=-v_-(D)+w(D)$ and $b(D)=v_+(D)+w(D)$.

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arXiv2022

The ropelength conjecture of alternating knots

A long standing conjecture states that the ropelength of any alternating knot is at least proportional to its crossing number. In this paper we prove that this conjecture is true. That is, there exists a constant $b_0>0$ such that $R(K)\ge b_0Cr(K)$ for any alternating knot $K$, where $R(K)$ is the ropelength of $K$ and $Cr(K)$ is the crossing number of $K$. In this paper, we prove that this conjecture is true.

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arXiv2022

The average genus of oriented rational links with a given crossing number

In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of oriented rational knots and links with any given crossing number.

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arXiv2021

Knots with equal bridge index and braid index

In this paper we are interested in BB knots, namely knots and links where the bridge index equals the braid index. Supported by observations from experiments, it is conjectured that BB knots possess a special geometric/physical property (and might even be characterized by it): if the knot is realized by a (closed) springy metal wire, then the equilibrium state of the wire is in an almost planar configuration of multiple (overlapping) circles. In this paper we provide a heuristic explanation to the conjecture and explore the plausibility of the conjecture numerically. We also identify BB knots among various knot families. For example, we are able to identify all BB knots in the family of alternating Montesinos knots, as well as some BB knots in the family of the non-alternating Montesinos knots, and more generally in the family of the Conway algebraic knots. The BB knots we identified in the knot families we considered include all of the 182 one component BB knots with crossing number up to 12. Furthermore, we show that the number of BB knots with a given crossing number $n$ grows exponentially with $n$.

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arXiv2021

The Ropelength of Special Alternating Knots

A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the ropelength of the link from below. Thus in the case an alternating link has a maximum braid index proportional to its minimum crossing number, such as the $T(2,2n)$ torus link, then the ropelength of the link is bounded below by a constant multiple of its minimum crossing number. However if the maximum braid index of a link is small compared to its crossing number, then there are no known results about whether its ropelength is bounded below by a constant multiple of its crossing number. For example, the $T(2,2n+1)$ torus knot has a minimum knot diagram that looks almost identical to that of the $T(2,2n)$ torus link, yet whether its ropelength is bounded below by a constant multiple of $n$ remains open to date. In this paper, we provide a first such result, and in fact for a large class of alternating knots. Specifically, we prove that if an alternating knot (namely a link with only one component) has a reduced alternating knot diagram in which the crossings are either all positive or all negative (such a knot is called a special alternating knot), then its ropelength is bounded below by a constant multiple of its crossing number.

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arXiv2020

Gabor single-frame and multi-frame multipliers in any given dimension

Functional Gabor single-frame or multi-frame generator multipliers are the matrices of function entries that preserve Parseval Gabor single-frame or multi-frame generators. An interesting and natural question is how to characterize all such multipliers. This question has been answered for several special cases including the case of single-frame generators in two dimensions and the case of multi-frame generators in one-dimension. In this paper we completely characterize multipliers for Gabor single-frame and multi-frame generators with respect to separable time-frequency lattices in any given dimension. Our approach is general and applies to the previously known cases as well.

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arXiv2020

Writhe-like invariants of alternating links

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are "writhe-like" invariants in the sense that they are also link invariants among reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

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arXiv2020

Gabor Functional Multiplier in the Higher Dimensions

For two given full-rank lattices $\mathcal{L}=A\mathbb{Z}^d$ and $\mathcal{K}=B\mathbb{Z}^d$ in $\mathbf{R}^d$, where $A$ and $B$ are nonsingular real $d\times d$ matrices, a function $g(\bf{t})\in L^2(\mathbf{R}^d)$ is called a Parseval Gabor frame generator if $\sum_{\bf{l},\bf{k}\in\mathbb{Z}^d}|\langle f, {e^{2πi\langle B\bf{k},\bf{t}\rangle}}g(\bf{t}-A\bf{l})\rangle|^2=\|f\|^2$ holds for any $f(\bf{t})\in L^2(\mathbf{R}^d)$. It is known that Parseval Gabor frame generators exist if and only if $|\det(AB)|\le 1$. A function $h\in L^{\infty}(\mathbf{R}^d)$ is called a functional Gabor frame multiplier if it has the property that $hg$ is a Parseval Gabor frame generator for $L^2(\mathbf{R}^d)$ whenever $g$ is. It is conjectured that an if and only if condition for a function $h\in L^{\infty}(\mathbf{R}^d)$ to be a functional Gabor frame multiplier is that $h$ must be unimodular and $h(\bf{x})\overline{h(\bf{x}-(B^T)^{-1}\bf{k})}=h(\bf{x}-A\bf{l})\overline{h(\bf{x}-A\bf{l}-(B^T)^{-1}\bf{k})},\ \forall\ \bf{x}\in \mathbf{R}^d$ {\em a.e.} for any $\bf{l},\bf{k}\in \mathbb{Z}^d$, $\bf{k}\not=\bf{0}$. The if part of this conjecture is true and can be proven easily, however the only if part of the conjecture has only been proven in the one dimensional case to this date. In this paper we prove that the only if part of the conjecture holds in the two dimensional case.

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arXiv2020

The number of oriented rational links with a given deficiency number

Let $U_n$ be the set of un-oriented and rational links with crossing number $n$, a precise formula for $|U_n|$ was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let $Λ_n$ be the set of oriented rational links with crossing number $n$ and let $Λ_n(d)$ be the set of oriented rational links with crossing number $n$ ($n\ge 2$) and deficiency $d$. In this paper, we derive precise formulas for $|Λ_n|$ and $|Λ_n(d)|$ for any given $n$ and $d$ and show that $$ Λ_n(d)=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor}, $$ where $F_n^{(d)}$ is the convolved Fibonacci sequence.

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arXiv2019

Invariants of rational links represented by reduced alternating diagrams

A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called {\em all-even form}). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced alternating form, or equivalently the nonalternating form of the corresponding rational number.

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arXiv2019

Braid Index Bounds Ropelength From Below

For an un-oriented link $\mathcal{K}$, let $L(\mathcal{K})$ be the ropelength of $\mathcal{K}$. It is known that when $\mathcal{K}$ has more than one component, different orientations of the components of $\mathcal{K}$ may result in different braid index. We define the largest braid index among all braid indices corresponding to all possible orientation assignments of $\mathcal{K}$ the {\em absolute braid index} of $\mathcal{K}$ and denote it by $\textbf{B}(\mathcal{K})$. In this paper, we show that there exists a constant $a>0$ such that $L(\mathcal{K})\ge a \textbf{B}(\mathcal{K}) $ for any $\mathcal{K}$, {\em i.e.}, the ropelength of any link is bounded below by its absolute braid index (up to a constant factor).

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arXiv2019

A Diagrammatic Approach for Determining the Braid Index of Alternating Links

It is well known that the braid index of a link equals the minimum number of Seifert circles among all link diagrams representing it. For a link with a reduced alternating diagram $D$, $s(D)$, the number of Seifert circles in $D$, equals the braid index $\textbf{b}(D)$ of $D$ if $D$ contains no {\em lone crossings} (a crossing in $D$ is called a {\em lone crossing} if it is the only crossing between two Seifert circles in $D$). If $D$ contains lone crossings, then $\textbf{b}(D)$ is strictly less than $s(D)$. However in general it is not known how $s(D)$ is related to $\textbf{b}(D)$. In this paper, we derive explicit formulas for many alternating links based on any minimum projections of these links. As an application of our results, we are able to determine the braid index for any alternating Montesinos link explicitly (which include all rational links and all alternating pretzel links).

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arXiv2018

The Braid Index Of Reduced Alternating Links

It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Murasugi had conjectured that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link. This conjecture turned out to be false. In this paper we prove the next best thing that one could hope for: we characterize exactly those alternating links for which their braid indices equal to the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if $D$ is a reduced alternating link diagram of an alternating link $L$, then $b(L)$, the braid index of $L$, equals the number of Seifert circles in $D$ if and only if $G_S(D)$ contains no edges of weight one. Here $G_S(D)$, called the Seifert graph of $D$, is an edge weighted simple graph obtained from $D$ by identifying each Seifert circle of $D$ as a vertex of $G_S(D)$ such that two vertices in $G_S(D)$ are connected by an edge if and only if the two corresponding Seifert circles share crossings between them in $D$ and that the weight of the edge is the number of crossings between the two Seifert circles. This result is partly based on the well known MFW inequality, which states that the $a$-span of the HOMFLY polynomial of $L$ is a lower bound of $2b(L)-2$, as well as a result due to Yamada, which states that the minimum number of Seifert circles over all link diagrams of $L$ equals $b(L)$.

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arXiv2016

The HOMFLY Polynomial of Links in Closed Braid Form

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an $n$-string closed braid diagram that is also reduced and alternating, is exactly $n$. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

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