Knot invariants derived from character theory

Knot invariants play a central role at various places in mathematics and physics. Furthermore, the classification of knots is still an open mathematical problem. Therefore new approaches to knot invariants are always desirable. We develop knot invariants from the character Hopf algebras of centralizer subgroups of the $GL(n)$ groups in the stable limit $n\to \mathrm{\infty }$. This invariants are induced by plethystic branchings. Usually one exploits the homomorphisms $G\text{righarrow}U(SL(2))$ or $G\to U_q(SL(2))$ to derive e.g. the Jones polynomial. Our method draws invariants directly from the character rings of infinitely many centralizer subgroups of $GL(\mathrm{\infty })$ and does not need mandatorily a $q$-deformation. We show how state models can be derived from our method and how it related for example to the Kauffman bracket and Jones polynomial.

Joint work with: Peter D. Jarvis, Hobart, and Ronald C. King, Southampton.