Abstract. We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

Abstract. For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are precisely two unknotting crossing changes. The proof uses sutured manifold theory and an analysis of the arc complex of the once-punctured torus.Mathematics Subject Classification (2010). 57M25, 57N10.

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type. 57M25; 57M27It is one of the founding theorems of knot theory that any two diagrams of a given link may be changed from one into the other by a sequence of Reidemeister moves. One of the reasons why this result is so crucial to the subject is that it allows one to define a link invariant as an invariant of a diagram which is unchanged under Reidemeister moves. In this paper we shall address the question of whether, given any two diagrams of a knot or link, there exists a sequence of Reidemeister moves between them which is sorted by type. We answer this question in the affirmative with the following theorem:Theorem 1 Given two diagrams D 1 and D 2 for a link L, D 1 may be turned into D 2 by a sequence of " 1 moves, followed by a sequence of " 2 moves, followed by a sequence of 3 moves, followed by sequence of # 2 moves. Furthermore, if D 1 and D 2 are diagrams of a link where the winding number and framing of each component is the same in each diagram, then D 1 may be turned into D 2 by a sequence of " 2 moves, followed by a sequence of 3 moves, followed by a sequence of # 2 moves.

Abstract. We present a new, practical algorithm to test whether a knot complement contains a closed essential surface. This property has important theoretical and algorithmic consequences; however, systematically testing it has until now been infeasibly slow, and current techniques only apply to specific families of knots. As a testament to its practicality, we run the algorithm over a comprehensive body of 2979 knots, including the two 20-crossing dodecahedral knots, yielding results that were not previously known.The algorithm derives from the original Jaco-Oertel framework, involves both enumeration and optimisation procedures, and combines several techniques from normal surface theory. This represents substantial progress in the practical implementation of normal surface theory, in that we can systematically solve a theoretically double exponential-time problem for significant inputs. Our methods are relevant for other difficult computational problems in 3-manifold theory, ranging from testing for Haken-ness to the recognition problem for knots, links and 3-manifolds.

Physical knots and links are one-dimensional submanifolds of ‫ޒ‬ 3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian split link, a two-component link that is split in the classical theory but cannot be split with a physical isotopy.