Primality of theta-curves with proper rational tangle unknotting number one

The Montesinos trick for proper rational tangle replacement

Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper rational tangle replacement and use it to study knots with proper rational unknotting number one. We prove that knots with proper rational unknotting number one are prime and classify the alternating knots with proper rational unknotting number one. We also study Montesinos knots with proper rational unknotting number one.

H(2)-Unknotting Operation Related to 2-Bridge Links

This paper concerns the H(2)-unknotting numbers of links related to 2-bridge links. It consists of three parts. In the first part, we consider a necessary and sufficient condition for a 2-bridge link to have H(2)-unknotting number one. The second part concerns an explicit form of composite links with H(2)-unknotting number one. In the last part, we develop a method of studying the H(2)-unknotting numbers of some tangle unknotting number one knots via 2-bridge knots.

Khovanov 同调与有理 unknotting

Khovanov homology and rational unknotting

本文提出一个基于 Khovanov 同调的新不变量 $λ$，用于下界估计有理 unknotting 数，并构造了具有任意非负整数值的结示例。

Levelling an unknotting tunnel

It is a consequence of theorems of Gordon-Reid [Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Ramifications, 4 (1995) 389-409] and Thompson [Thin position and bridge number for knots in the 3-sphere, Topology, 36 (1997) 505-507] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [A note on unknotting tunnels for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku U…

Rational tangle replacements and knot Floer homology

From the link Floer complex of a link $K$, we extract a lower bound $t_q'(K)$ for the rational unknotting number of $K$ (i.e. the minimum number of rational replacements required to unknot $K$). Moreover, we show that the torsion obstruction $t_q(K)=\hat{t}(K)$ from an earlier paper of Alishahi and the author is a lower bound for the proper rational unknotting number. Moreover, $t_q(K\#K')=\max\{t_q(K),t_q(K')\}$ and $t'_q(K\#K')=\max\{t'_q(K),t'_q(K')\}$. For the torus knot $K=T_{p,pk+1}$ we compute $t'_q(K)=\lfloor p/2\rfloor$ and $t_q(K)=p-1$.

Knots with unknotting number 1 and essential Conway spheres

For a knot K in S^3, let T(K) be the characteristic toric sub-orbifold of the orbifold (S^3,K) as defined by Bonahon and Siebenmann. If K has unknotting number one, we show that an unknotting arc for K can always be found which is disjoint from T(K), unless either K is an EM-knot (of Eudave-Munoz) or (S^3,K) contains an EM-tangle after cutting along T(K). As a consequence, we describe exactly which large algebraic knots (ie algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determin…