Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Knot adjacency and satellites   A knot K is called n-adjacent to the unknot, if K admits a projection
containing n generalized crossings such that changing any m (no larger than n)
of them yields a projection of the unknot. We show that a non-trivial satellite
knot K is n-adjacent to the unknot, for some n>0, if and only if it is
n-adjacent to the unknot in any companion solid torus. In particular, every
model knot of K is n-adjacent to the unknot. Along the way of proving these
results, we also show that 2-bridge knots of the form K_{p/q}, where
p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have
genus one and are 2-adjacent to the unknot.

23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg