Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the intersection of unknotting tunnels and the decomposing annulus in
  connected sums   Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot
$K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up
to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is
a minimal collection of disks we study the components of $V_1-N(A)$ and show
that at most one component is a 3-ball meeting $A$ in two disks. This is a
crucial step in proving the conjecture that a necessary and sufficient
condition for the tunnel number of a connected sum to be less than or equal to
the sum of the tunnel numbers is that one of the knots has a Heegaard splitting
in which a merdian curve is primitive.

37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg