Colloquium Mathematics - Dr. Takefumi Nosaka, Institute of Science Tokyo

Title: Yang–Baxter co-Colorings of braids and link invariants of Groebner basis type

Abstract:

We study link invariants derived from set-theoretic Yang–Baxter operators $(X,R)$. In previous knot theory, there are many studies of braid colorings. In contrast, given a Markov-stable Yang–Baxter datum $(X,R)$, we associate to a braid $\beta \in B_n$ a co-coloring module defined as a cokernel of $\beta - id_{X^n}$. Markov stabilization acts by adjoining a free summand, so suitable shifts of Fitting ideals (or Groebner basis) yield invariants of braid closures. Examples indicate whether these invariants can distinguish links beyond Alexander-type data and are amenable to explicit computation.