Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On keen weakly reducible bridge spheres   A bridge sphere is said to be keen weakly reducible if it admits a unique
pair of disjoint compressing disks on opposite sides. In particular, such a
bridge sphere is weakly reducible, not perturbed, and not topologically minimal
in the sense of David Bachman. In terms of Jennifer Schultens' width complex, a
link in bridge position with respect to a keen weakly reducible bridge sphere
is distance one away from a local minimum. In this paper, we give infinitely
many examples of keen weakly reducible bridge spheres for links in $b$-bridge
position for $b \geq 3$. As a corollary, we see that topological minimality is
not necessarily preserved under the connected sum operation.

20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
1
The categories of this paper are: cs