Narrow Results

We prove a simple necessary and sufficient condition for a two-bridge knot $K(p,q)$ to be quasipositive, based on the continued fraction expansion of $p/q$. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in Appendix A, by Stepan Orevkov.

The Kauffman–Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize this conjecture by stating it in terms of homology of the double cover of S³. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman–Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman–Harary conjecture for all Montesinos links. Finally, we discuss on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S³ along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. -moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.

In this paper, we show that Montesinos links of the form $L(1/2,1/2,p/q;e)$, which we call $(2,2,r)$-Montesinos links, have finite involutory quandles. This generalizes an observation of Winker regarding the $(2,2,q)$-pretzel links. We also describe some properties of these quandles.

Given a link $L$, a representation ${\pi }_1(S^3-L)\to \mathrm{\text{SL}}(2,ℂ)$ is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when $L$ is a Montesinos link.

We give a simple condition for the existence of a nontrivial quandle coloring on a Montesinos link, which describes the distribution of the zeros of the Alexander polynomial. By this condition, we prove that the real parts of the zeros of the Alexander polynomial of any link that admits an alternating pretzel diagram are greater than $-1$: Hoste’s conjecture holds for such a link. Furthermore, we show the existence of infinitely many counterexamples for Hoste’s conjecture.

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 $L$ be a Montesinos link $M(-p,q,r)$ with positive rational numbers $p,q$ and $r$, each less than 1, and $c(L)$ the minimal crossing number of $L$. Herein, we construct arc presentations of $L$ with $c(L)$, $c(L)-1$ and $c(L)-2$ arcs under some conditions for $p$, $q$ and $r$. Furthermore, we determine the arc index of infinitely many Montesinos links.

This paper is the second installment of our in-depth study on the braid index problem of pretzel links. We classify pretzel links into three categories based on the Seifert circle decompositions in their standard link diagrams. In our initial paper, we successfully identified the braid indices for all Type 1 and Type 2 pretzel links. In this paper, we share our findings on Type 3 pretzel links. We are able to determine the exact braid indices for most Type 3 pretzel links. For the small subset of Type 3 pretzel links where exact braid indices remain undetermined, we establish their braid indices within a range of two consecutive integers. With some numerical evidence, we conjecture that in such an exceptional case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas.

In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. Pretzel links can be divided into three types according to the Seifert circle decompositions of their standard link diagrams. In this paper, we present our results on Type 1 and Type 2 pretzel links, namely the complete formulation of the braid indices for all non-alternating Type 1 and Type 2 pretzel links. 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.