Carl D. Modes, Marcelo O. Magnasco
Early last century witnessed both the complete classification of 2-dimensional manifolds and a proof that classification of 4- dimensional manifolds is undecidable, setting up 3-dimensional manifolds as a central battleground of topology to this day. A rather important subset of the 3-manifolds has turned out to be the knotspaces, the manifolds left when a thin tube around a knot in 3D space is excised. Tesselating the knotspace of arbitrary knots into polyhedral complexes is a fundamental step in knot computational topology, yet it has been hitherto carried out using ad hoc methods of uncontrolled computational complexity. Here we introduce a geometrically-inspired template for the lower-dimensional deformation retract of the knotspace of arbitrary knots and links in 3-space. The template can be constructed directly from a planar presentation of the knot with C crossings using at most 12C polygons bounded by 64C edges, in time O(C). We show the utility of our template by deriving a novel presentation of the fundamental group, from which we motivate a measure of complexity of the knot diagram.
View original:
http://arxiv.org/abs/1302.1146
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