Abstract:
Invariants for knots and links typically can be characterized in one of two ways. There are invariants which are easy to define yet are difficult to compute and there are invariants which allow for more straightforward computation but usually have involved definitions. Quantum invariants for knots and links belong to this second category of invariants. However, the quantum invariants associated to the Lie algebras $\mathfrak{s}\mathfrak{l}_2$ and $\mathfrak{s}\mathfrak{l}_3$ can be defined using skein relations, which allows for a simple definition that side steps an explicit mention of quantum groups and their representations. This allows the $\mathfrak{s}\mathfrak{l}_2$ and $\mathfrak{s}\mathfrak{l}_3$ quantum invariants to be defined in a simple manner, but also allowing for easy computation. In this talk, I will define the $\mathfrak{s}\mathfrak{l}_2$ and $\mathfrak{s}\mathfrak{l}_3$ quantum invariants in terms of skein relations and compute a few examples. I will further give examples showing that there are nonisotopic links that are not distinguished by $\mathfrak{s}\mathfrak{l}_2$ invariant but are distinguished by the $\mathfrak{s}\mathfrak{l}_3$ invariant. This talk is intended for a general math audience.