A tale of two tangles

May 24, 2023
A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. We will discuss two instances in which tangle decompositions can be used to address classic problems in knot theory. In the first instance, we will use a very simple tangle decomposition to prove an equivariant version of the cosmetic surgery conjecture. The proof strategy relies on a reinterpretation of Khovanov homology and Bar-Natan's tangle invariant in terms of immersed curves on the four-punctured sphere. The second instance involves a complicated tangle, the role of which is to generalize the statement 'unknotting number one knots are prime' to spatial theta graphs. This spans joint work with two different sets of authors.