Unknotting operations on knots and links

Abstract

By considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu, Knot theory and game (in Japanese) (Asakura Publishing, Tokyo, 2013); L. Rudolph, Qualitative Mathematics for the Social Sciences (Routledge, London, 2013); K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kupita (Birkhauser, Boston, MA, 1996)], etc.). In this paper, we see how knots can be unknotted (and therefore how they are knotted) by considering various unknotting operations and their associated unknotting numbers.