Abstract: A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. In this talk, we'll discuss how Heegaard Floer and Khovanov homologies can be approached and calculated using tangle decompositions. In both cases, the algebraic invariants can be realized geometrically as immersed curves on the four-punctured sphere. This strategy turns out to be quite useful for studying L-space knots and investigating two classic open problems: the cosmetic surgery conjecture and the cosmetic crossing conjecture. This is joint with Kotelskiy, Lidman, Watson and Zibrowius.
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