Journal of Knot Theory and Its Ramifications

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\mathrm{\Delta }(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\mathrm{\Delta }(x,y)$ to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then $\mathrm{\Delta }(-1,0)=1$. If the standard form of the rational link has $M$ monochromatic twist sites, and the $j$th monochromatic twist site has $m_j$ crossings, then $\mathrm{\Delta }(-1,0)={\prod }_{j=1}^M(m_j+1)$. Our proof employs Kauffman’s clock moves and a lattice for the terms of $\mathrm{\Delta }(x,y)$ in which the $y$-power cannot decrease.

We show that the 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ is an L-space and its fundamental group is not left-orderable. Therefore, the family of 3-fold cyclic branched cover of any genus 2 two-bridge knot $K_{[-2q,2s,-2t,2l]}$ verifies the $L$-space conjecture. We also show that if $K_{[2k,-2l]}$ is a two-bridge knot with $k\ge 2$, $l>0$, then the fundamental group of the 5-fold cyclic branched cover of $K_{[2k,-2l]}$ is not left-orderable, which will complete the proof that the fundamental group of the 5-fold cyclic branched cover of any genus 1 two-bridge knot is not left-orderable.

In this paper, we define transcendental polynomial invariants for two-component virtual string links. One of these invariants is a strictly refinement of the linking polynomial in [M. Xu and H. Gao, Linking polynomials of virtual string links, Sci. China Math. 61(7) (2018) 1287–1302]. It is a homotopy invariant and can distinguish some virtual string links from their mirror images. We also define a transcendental polynomial invariant for two-component flat virtual string links. These invariants can be used to study the periodicity and linking crossing number of virtual string links.

A twisted torus knot $T(p,q,r,s)$ is obtained from a torus knot $T(p,q)$ by twisting $r$ adjacent strands of $T(p,q)$ fully $s$ times. In this paper, we determine the parameters $p,q,r,s$ for which $T(p,q,r,s)$ is a torus knot with $s>0$.

We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states adding an even number of crossings to a twist region will not change whether the knots are perfectly straight or not perfectly straight.

We compute Cayley graphs and automorphism groups for all finite $n$-quandles of two-bridge and torus knots and links, as well as torus links with an axis.