For a knot K let c K denote the crossing number of the knot

For a knot K let c (K) denote the crossing number of the knot (that is the minimum number of crossings over all the knot diagrams of K). Show that for any knots K, K\' we have c (K # K\') lessthanorequalto c(K) + c(K\'), where K # K denotes the connect sum of K and K\'.

Solution

Solution :- We have to prove that c(K#K\') c(K)+c(K\')

More generally, one can speculate about the crossing number of satellite knots.
Here, there are a variety of conjectures, all of which remain wide open at present. The
simplest of these asserts that the crossing number of a non-trivial satellite knot is at
least the crossing number of its companion. To explain this, we fix some terminology.
A knot K is a non-trivial satellite knot with companion knot L if K lies in a regular
neighbourhood N(L) of the non-trivial knot L, and K does not lie in a 3-ball contained
in N(L), and K is not a core curve of the solid torus N(L).