Narrow Results

We explain progress in computing the cabled Jones, HOMFLY and Kauffman polynomial. This is applied, first, in combination with some group theoretic considerations, to the tabulation of low-crossing mutants. Then we study the distinction of mutants, with particular regard to the symmetric mutants. We discuss the determination of braid index as another application of our computational methods.

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.

We give an upper bound for the dealternating number of a closed 3-braid. As applications, we determine the dealternating numbers, the alternation numbers and the Turaev genera of some closed positive 3-braids. We also show that there exist infinitely many positive knots with any dealternating number (or any alternation number) and any braid index.

We prove that the Murasugi–Przytycki index of the link graph determines an upper bound for the number of reducing operations that can be performed on a link diagram to reduce the number of Seifert circles.

We consider a surface link in the 4-space which can be presented by a simple branched covering over the standard torus, which we call a torus-covering link. Torus-covering links include spun T²-knots and turned spun T²-knots. In this paper we braid a torus-covering link over the standard 2-sphere. This gives an upper estimate of the braid index of a torus-covering link. In particular we show that the turned spun T²-knot of the torus (2, p)-knot has the braid index four.

The zeroth coefficient polynomial is a one variable polynomial contained in the HOMFLYPT polynomial. In this paper, we give a basic computation of the zeroth coefficient polynomial of a 2-cable knot. In particular, we compute the zeroth coefficient polynomials of the 2-cable knots of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov–Rozansky homology. As a result, we distinguish the Kanenobu knots completely. Moreover, we estimate the braid indices of the Kanenobu knots.

In our previous paper, we studied the braid index of the Kanenobu knot k(n) for n ≥ 0. In this paper, we study the braid index of the Kanenobu knot K(a, b) for a, b ∈ ℤ. In particular, k(n) is K(2n, -2n). The MFW inequality is known for giving a lower bound of the braid index of an oriented link by applying the HOMFLYPT polynomial. The HOMFLYPT polynomial of K(a, b) is given by Professor Taizo Kanenobu. Therefore, we have a lower bound of the braid index of K(a, b). The purpose of this paper is to give an upper bound of the braid index of K(a, b). As a result, we determine the braid indices of infinitely many Kanenobu knots.

In this paper, we calculate the Kauffman polynomials $F(K(p,q);a,z)$ of Kanenobu knots $K(p,q)$ with $p,q$ half twists and determine their spans on the variable $a$ completely. As an application, we determine the arc index of infinitely many Kanenobu knots. In particular, we give sharper lower bounds of the arc index of $K(2n,-2n)$ by using canonical cabling algorithm and the 2-cable $\mathrm{\Gamma }$-polynomials. Moreover, we give sharper upper bounds of the arc index of some Kanenobu knots by using their braid presentations.

For each positive integer $n$ we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to $n$, braid index equal to $n+3$ and Turaev genus equal to $n$.

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index $b(L)$ and arc index $\alpha (L)$ for any knot or non-split link $L$, which are $b(L)\le \frac{1}{2}c(L)+1$ and $\alpha (L)\le c(L)+2$. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of $L$.

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and the double crossing braid index for unoriented links, namely ${\beta }_2(L)\le c_3(L)+1$. This yields a new method for determining braid indices. We find an infinite family of knots that achieve equality, which allows us to determine both the double crossing braid index and the triple crossing number of these knots.

For an unoriented link ${𝒦}$, let $L({𝒦})$ be the ropelength of ${𝒦}$. It is known that in general $L({𝒦})$ is at least of the order $O({(\mathrm{\text{Cr}}({𝒦}))}^{3/4})$, and at most of the order $O(\mathrm{\text{Cr}}({𝒦}){ln}^5(\mathrm{\text{Cr}}({𝒦}))$ where $\mathrm{\text{Cr}}({𝒦})$ is the minimum crossing number of ${𝒦}$. Furthermore, it is known that there exist families of (infinitely many) links with the property $L({𝒦})=O(\mathrm{\text{Cr}}({𝒦}))$. A long standing open conjecture states that if ${𝒦}$ is alternating, then $L({𝒦})$ is at least of the order $O(\mathrm{\text{Cr}}({𝒦}))$. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant $a>0$ such that $L({𝒦})\ge aB({𝒦})$ for any ${𝒦}$, where $B({𝒦})$ is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of ${𝒦}$ (called the maximum braid index of ${𝒦}$). Consequently, $L({𝒦})\ge O(\mathrm{\text{Cr}}({𝒦}))$ for any link ${𝒦}$ whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram $D$, it is known that $s(D)$, the number of Seifert circles in $D$, equals the braid index $b(D)$ of $D$ if $D$ contains no lone crossings, where a crossing in $D$ is called a lone crossing if it is the only crossing between two Seifert circles in $D$. If $D$ contains lone crossings, then one can reduce the number of Seifert circles in $D$ using link-type preserving moves such as the Murasugi–Przytycki operation. Let $r(D)\ge 1$ be the maximum number of Seifert circles in $D$ that can be reduced, then we have $b(D)\le s(D)-r(D)$. On the other hand, if the $a$-span of the HOMFLY polynomial of $D$ satisfies the equality $a$-span$/2+1=s(D)-r(D)$, then the Morton–Frank–Williams (MFW) inequality $a$-span$/2+1\le b(D)$ leads us to the simple braid index formula $b(D)=s(D)-r(D)$. In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality $a$-span$/2+1=s(D)-r(D)$. Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.

Let ${{𝒰}}_n$ be the set of un-oriented and rational links with crossing number $n$, a precise formula for $\left|{{𝒰}}_n\right|$ was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let ${\mathrm{\Lambda }}_n$ be the set of oriented rational links with crossing number $n$ and let ${\mathrm{\Lambda }}_n(d)$ 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 $\left|{\mathrm{\Lambda }}_n\right|$ and $|{\mathrm{\Lambda }}_n(d)|$ for any given $n$ and $d$ and show that

In this paper, we are interested in BB knots, namely knots and links whose bridge index and braid index are equal. 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 $n$.

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, Rachel, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. We prove that all 2-bridge knots and links with a braid index greater than three are curly. We also show that many alternating oriented Montesinos links are curly.

A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links.

In this work, we find a closed form formula for the braid index of an $n$-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary, effective, and self-contained, and partially recovers work of Birman–Kofman. Along the way, we show that the disparate definitions of twisted torus knots in the literature agree.

In this paper, we consider two properties on the braid index of a two-bridge knot. We prove an inequality on the braid indices of two-bridge knots if there exists an epimorphism between their knot groups. Moreover, we provide the average braid index of all two-bridge knots with a given crossing number.

Let $\alpha$ be a map from the set of all knot types ${𝒦}$ to a set $X$. Let $\beta$ be a map from ${𝒦}$ to a set $Y$. We define the relation between $\alpha$ and $\beta$ to be the image of a map $(\alpha ,\beta )$ from ${𝒦}$ to $X\times Y$ sending an element $K$ of ${𝒦}$ to $(\alpha (K),\beta (K))$. We determine the relations between $\alpha$ and $\beta$ for certain $\alpha$ and $\beta$ such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus. This is a study of geography problem in knot theory.