Journal of Knot Theory and Its Ramifications

We introduce the notion of bicolored diagrams which are closely related to the region crossing changes. Moreover, we refine Cheng’s results on the region crossing changes and propose a certain way to calculate the Arf invariant of a proper link using a bicolored diagram.

We use the Bar-Natan Ж-correspondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Ж-map is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in $S^4$) is functorial under concordance. In addition, we extend classical results of Chen, Milnor and Hillman on the lower central series of link groups to links in thickened surfaces. Our main result is that the generalized Alexander polynomial vanishes on any knot in a thickened surface which is virtually concordant to a homologically trivial knot. In particular, this shows that it vanishes on virtually slice knots. We apply it to complete the calculation of the slice genus for virtual knots with four crossings and to determine non-sliceness for a number of 5-crossing and 6-crossing virtual knots.

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich–Zagier function and the generating function for strongly unimodal sequences.

Let $M^{\mathrm{\text{reg}}}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n>1$ let $M_n^{\mathrm{\text{reg}}}$ be the moduli space of all $n$-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$. We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M_n^{\mathrm{\text{reg}}}$ by a very simple formula. This 1-cocycle depends on a natural number $a\in \mathbb{Z}\cong H_1(V^3;\mathbb{Z})$ with $0<a<n$ as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on $H_0(M_n^{\mathrm{\text{reg}}})\times H_0(M^{\mathrm{\text{reg}}})$ already for $n=2$.

Given an oriented minimal genus Seifert surface $F^{\prime }$ for a $(1,1)$-knot $K$ it is possible to surger $F^{\prime }$ along annuli to obtain a simple minimal Seifert surface$F$ . Such a surface can be put in a very nice position with respect to the $(1,1)$-position of the knot $K$. Using this kind of surfaces we give a description of a $(1,1)$-knot of genus $g$ as a vertical banding of $(1,1)$-knots of genus smaller than $g$. In addition, we show that any rational knot of genus $g$ is obtained as a vertical banding of $g$ genus one rational knots.

We calculate the Ohtsuki invariants ${\lambda }_i(M)$ $(i=0,1,2,3)$ of every Brieskorn–Hamm manifold $M$ which is a rational homology $3$-sphere. We denote the order of $H_1(M;\mathbb{Z})$ by $H$. By the result, we show that the third Ohtsuki invariant ${\lambda }_3(M)$ of the Brieskorn–Hamm manifolds $M$ with $H=1$ is negative, and the third Ohtsuki invariant ${\lambda }_3(M)$ of most Brieskorn–Hamm manifolds $M$ with $H\ge 2$ is positive.

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.

The Gordian graph and H(2)-Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)-moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial, and an invariant related to tricolorability. We show, in all cases considered, that the quotient graphs are Gromov hyperbolic. We then prove a collection of results about the graph isomorphism type of the quotient graphs. In particular, we find that the H(2)-Gordian graph of links modulo the relation induced by the span of the Jones polynomial is isomorphic with the complete graph on infinitely many vertices.