Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Construction and decomposition of knots as Murasugi sums of Seifert
  surfaces   A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy
classes of knots. This operation endows $\mathcal{S}$ with a directed graph
structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots
in $M\kern-1pt SG(K)$ has the structure of a bi-directed complete graph, which
is not the case if we restrict the complexity of Murasugi sums. For that
purpose, we show that any knot is a Murasugi sum of any two knots, and we give
lower and upper bounds for the minimal complexity of Murasugi sum to obtain
$K_3$ by $K_1$ and $K_2$. As an application, we show that given any three
knots, there is a braid for one knot which splits along a string into braids
for the other two knots.

33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc