The Kauffman Skein Algebra of the Torus

We give a presentation of the Kauffman (BMW) skein algebra of the torus. This algebra is the “type $BCD$” analogue of the Homflypt skein algebra of torus, which was computed in earlier work of the 1st and 3rd authors [17]. This suggests the existence of a “type $BCD$” version of the Hall algebra of an elliptic curve [4]. In the appendix, we show this presentation is compatible with the Frohman–Gelca description of the Kauffman bracket (Temperley–Lieb) skein algebra of the torus [12].