Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Geometric Knotspace Template   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. Given a knot diagram it would be
desirable to provide as compact a description of its knotspace as feasible;
hitherto this has been done by computationally tessellating the knotspace of a
given knot into polyhedral complexes using ad hoc methods of uncontrolled
computational complexity. Here we present an extremely compact representation
of the knotspace obtainable directly from a knot diagram; more technically, an
explicit, geometrically-inspired polygonal tessellation of a deformation
retract of the knotspace of arbitrary knots and links. Our 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.

8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat