Narrow Results

Let $K$ be a link of Conway’s normal form $C(m)$, $m\ge 0$, or $C(m,n)$ with $mn>0$, and let $D$ be a trigonal diagram of $K.$ We show that it is possible to transform $D$ into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [Chebyshev diagrams for two-bridge knots, Geom. Dedicata150 (2010) 405–425; E. Brugallé, P.-V. Koseleff, D. Pecker, Untangling trigonal diagrams, to appear in J. Knot Theory and its Ramifications]. We also give a sharp lower bound for the lexicographic degree of any knot, using real polynomial curves properties.

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.

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.

Fox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the absolute values of the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa–Murasugi conjecture for two-bridge knots.

We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K\ge J$ if there exists an epimorphism $f:{\pi }_1(S^3-K)\to {\pi }_1(S^3-J)$. We prove that if $K$ and $J$ are 2-bridge knots and $K>J$, then $\gamma (K)\ge 3\gamma (J)-4$. We also classify all pairs $(K,J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ and $J$ are 2-bridge knots and $K>J$, then $g(K)\ge 3g(J)-1$, where $g(K)$ denotes the genus of the knot $K$.

Let $M$ be a $\mathbb{Q}$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\ge 1$ admits an interval of left orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.