7 research outputs found
Twisted Alexander polynomials on curves in character varieties of knot groups
For a fibered knot in the 3-sphere the twisted Alexander polynomial
associated to an SL(2,C)-character is known to be monic. It is conjectured that
for a nonfibered knot there is a curve component of the SL(2,C)-character
variety containing only finitely many characters whose twisted Alexander
polynomials are monic, i.e. finiteness of such characters detects fiberedness
of knots. In this paper we discuss the existence of a certain curve component
which relates to the conjecture when knots have nonmonic Alexander polynomials.
We also discuss the similar problem of detecting the knot genus.Comment: 13 pages, 1 figure; to appear in International Journal of Mathematic
Bounds for the genus of a normal surface
This paper gives sharp linear bounds on the genus of a normal surface in a
triangulated compact, orientable 3--manifold in terms of the quadrilaterals in
its cell decomposition---different bounds arise from varying hypotheses on the
surface or triangulation. Two applications of these bounds are given. First,
the minimal triangulations of the product of a closed surface and the closed
interval are determined. Second, an alternative approach to the realisation
problem using normal surface theory is shown to be less powerful than its dual
method using subcomplexes of polytopes.Comment: 38 pages, 25 figure
Computing closed essential surfaces in knot complements
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. Problems of this kind have a doubly-exponential time complexity; nevertheless, with our new algorithm we are able to solve it for a large and comprehensive class of 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. Copyright 2013 ACM