We will discuss how certain knot invariants can be homogenized to yield invariants for braids. The idea of this type of homogenization has been around for decades, for example in pioneering work by Gambaudo-Ghys on Levine-Tristram signatures.

By focusing on knot invariants stemming from Heegaard Floer theory, we find a new inequality relating the fractional Dehn twist coefficient with the 4-genus of knots, answering a question of Hubbard-Kawamuro-Ceren Kose-Martin-Plamenevskaya-Raoux-Truong-Turner.

In fact, with the tool of homogenization, this new inequality is readily seen to be one of a family of inequalities, the most famous of which is the slice Bennequin inequality established by Kronheimer-Mrowka and Rudolph.

The meta goal of the talk is to invite the audience to consider what homogenization yields for their favorite (Floer homotopy) invariants.