Title: Lie superalgebra generalizations of the Jaeger-Kauffman-Saleur invariant
Speaker: Micah Chrisman (The Ohio State University - Marion Campus)
Speaker's URL: https://math.osu.edu/people/chrisman.76
Abstract: Jaeger-Kauffman-Saleur (JKS) identified the Alexander polynomial with the $U_q(\mathfrak{gl}(1|1))$ quantum invariant of classical links and extended this to a 2-variable invariant of links in thickened surfaces. Here we generalize this story for every Lie superalgebra of type $\mathfrak{gl}(m|n)$. First, we define a $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev invariant for virtual tangles. When $m=n=1$, this recovers the Alexander polynomial of almost classical knots, as defined by Boden-Gaudreau-Harper-Nicas-White. Next, an extended $U_q(\mathfrak{gl}(m|n))$ Reshetikin-Turaev invariant of virtual tangles is obtained by applying the Bar-Natan Zh-construction. This is equivalent to the 2-variable JKS-invariant when $m=n=1$, but otherwise our invariants are new whenever $n>0$. Furthermore, in contrast with the classical case, the virtual and extended $U_q(\mathfrak{gl}(m|n))$ invariants are not entirely determined by the difference $m-n$. For example, the invariants from $U_q(\mathfrak{gl}(2|0))$ (i.e. the classical Jones polynomial) and $U_q(\mathfrak{gl}(3|1))$ are distinct, as are the extended invariants from $U_q(\mathfrak{gl}(1|1))$ and $U_q(\mathfrak{gl}(2|2))$. Further applications and conjectures based on calculations will be discussed. This is joint work (in progress) with Anup Poudel.
URL associated with Seminar: https://www.asc.ohio-state.edu/math/vqss/
