Related papers: Untwisting 3-strand torus knots

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact…

We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in $\mathbb{R}^3$ proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of…

Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two…

The ribbonlength Rib$(K)$ of a knot $K$ is the infimum of the ratio of the length of any flat knotted ribbon with core $K$ to its width. A twisted torus knot $T_{p,q;r,s}$ is obtained from the torus knot $T_{p,q}$ by twisting $r$ adjacent…

Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give…

Experimental work suggests that the Seifert genus of a knot grows linearly with respect to the crossing number of the knot. In this article, we use a billiard table model for $2$-bridge or rational knots to show that the average genus of a…

We use Ozsv\'ath, Stipsicz, and Szab\'o's Upsilon-invariant to provide bounds on cobordisms between knots that `contain full-twists'. In particular, we recover and generalize a classical consequence of the Morton-Franks-Williams inequality…

We classify the positive definite intersection forms that arise from smooth 4-manifolds with torsion-free homology bounded by positive integer surgeries on the right-handed trefoil. A similar, slightly less complete classification is given…

A knot in $S^3$ is topologically slice if it bounds a locally flat disk in $B^4$. A knot in $S^3$ is rationally slice if it bounds a smooth disk in a rational homology ball. We prove that the smooth concordance group of topologically and…

Let $M_K$ be the 2-fold branched cover of a knot $K in $S^3$. If $H_1(M_K) = {\bf Z}_3 \oplus {\bf Z}_{3^{2i}} \oplus G$ where 3 does not divide the order of $G$ then $K$ is not of order 4 in the concordance group. This obstruction detects…

We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible…

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof…

We provide explicit formulas for the integer-valued smooth concordance invariant $\upsilon(K) = \Upsilon_K(1)$ for every 3-braid knot $K$. We determine this invariant, which was defined by Ozsv\'ath, Stipsicz and Szab\'o, by constructing…

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is…

Suppose an orientation preserving action of a finite group $G$ on the closed surface $\Sigma_g$ of genus $g>1$ extends over the 3-torus $T^3$ for some embedding $\Sigma_g\subset T^3$. Then $|G|\le 12(g-1)$, and this upper bound $12(g-1)$…

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…

Freedman and Krushkal showed that if the surgery conjecture and the $s$-cobordism conjecture hold for all topological 4-manifolds, then every link with pairwise zero linking numbers is topologically round handle slice. Kim, Powell, and…

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a…

We show that the fundamental group of the $3$-manifold obtained by $\frac{p}{q}$-surgery along the $(n-2)$-twisted $(3,3m+2)$-torus knot, with $n,m \ge 1$, is not left-orderable if $\frac{p}{q} \ge 2n + 6m-3$ and is left-orderable if…

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac pq$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac pq$-surgery on $K_0$ via an orientation preserving homeomorphism,…