Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Converting homotopies to isotopies and dividing homotopies in half in an
  effective way   We prove two theorems about homotopies of curves on 2-dimensional Riemannian
manifolds. We show that, for any epsilon > 0, if two simple closed curves are
homotopic through curves of bounded length L, then they are also isotopic
through curves of length bounded by L + epsilon. If the manifold is orientable,
then for any epsilon > 0 we show that, if we can contract a curve gamma
traversed twice through curves of length bounded by L, then we can also
contract gamma through curves bounded in length by L + epsilon.
  Our method involves cutting curves at their self-intersection points and
reconnecting them in a prescribed way. We consider the space of all curves
obtained in this way from the original homotopy, and use a novel approach to
show that this space contains a path which yields the desired homotopy.

4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs