Computing closed essential surfaces in knot complements

Abstract

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.

3-manifold groups

“…See also [Thu79,Oe84,Ag03,BRT12] for more examples of non-Haken manifolds. Jaco-Oertel [JO84, Theorem 4.3] (see also [BCT12]) found an algorithm which can decide whether or not a given closed irreducible 3-manifold is Haken.…”

Twisted Alexander Polynomials on Curves in Character Varieties of Knot Groups

“…This construction is based on [29,Theorem 3.2]. One can also check that the knots 10 99 and 10 123 are not small in the list given in [2]. Theorem 4.4.…”

A New Approach to Crushing 3-Manifold Triangulations

“…In the last decade, however, there has been strong progress in the realm of practical, usable algorithms on 3-manifolds. For instance, there are now practical implementations of unknot recognition, 3-sphere recognition and orientable prime decomposition [1,3], and recently more complex algorithms such as testing for closed essential surfaces have become viable [7,8].…”

“…Although the full process of obtaining a 0-efficient triangulation requires worstcase exponential time, recent techniques based on combinatorial optimisation have made this extremely fast in a range of experimental settings [6,8]. A notable application of crushing has been in 3-sphere recognition: here the introduction of Jaco and Rubinstein's 0-efficiency techniques was a major turning point that made 3-sphere recognition practical to implement for the first time [5,16].…”