Signed Heights of Knotoids

A relation between the crossing number and the height of a knotoid

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knot…

Parity in Knotoids

This paper investigates the parity concept in knotoids in $S^2$ and in $\mathbb{R}^2$ in relation with virtual knots. We show that the virtual closure map is not surjective and give specific examples of virtual knots that are not in the image. We introduce a planar version of the parity bracket polynomial for knotoids in $\mathbb{R}^2$. By using the Nikonov/Manturov theorem on minimal diagrams of virtual knots we prove a conjecture of Turaev showing that minimal diagrams of knot-type knotoids have zero height.

关于纽结的邻域

On the neighborhood of knots

本文提出通过研究纽结嵌入的邻域及其纽结oid谱来区分纽结，证明邻域不变量能比直接应用于纽结本身区分更多纽结，包括某些突变纽结和平凡纽结检测问题。