Semi-isotopic knots

TL;DR

This paper introduces the concept of semi-isotopy between knots, showing that every knot can be continuously deformed into an unknot through a specific annulus-based process.

Contribution

It defines semi-isotopy for knots and proves that all knots are semi-isotopic to the unknot, expanding understanding of knot equivalence.

Findings

Every knot is semi-isotopic to an unknot

Semi-isotopy involves an annulus in 4D space connecting knots

The concept generalizes isotopy with a new deformation method

Abstract

A $\textit{knot}$ is a possibly wild simple closed curve in $S^3$. A knot $J$ is $\textit{semi-isotopic}$ to a knot $K$ if there is an annulus $A$ in $S^3\times[0,1]$ such that $A\cap(S^3\times\{0,1\})=\partial A=(J\times\{0\})\cup(K\times\{1\})$ and there is a homeomorphism $e:S^1\times[0,1)\rightarrow A-(K\times\{1\})$ such that $e(S^1\times\{t\})\subset S^3\times\{t\}$ for every $t\in[0,1)$. $\textbf{Theorem.}$ Every knot is semi-isotopic to an unknot.