DAHAs and skein theory (Q2041642)

The authors give a skein theoretic realization of the double affine Hecke algebra \(\ddot{H}_n\) of type \(\mathfrak{gl}_n\). It is shown that the skein algebra \(\mathrm{BSK}_n(T^2,*)\) in the punctured torus is isomorphic to \(\ddot{H}_n\). The skein algebra \(\mathrm{BSK}_n(T^2,*)\) is conjectured to be isomorphic to the HOMFLY skein algebra on the punctured torus. Thus, using the result of \textit{O.~Schiffmann} and \textit{E.~Vasserot} [Compos. Math. 147, No. 1, 188--234 (2011; Zbl 1234.20005)], the authors give a surjective map from the simple closed curves on the punctured torus to the generators of the elliptic Hall algebra of \textit{I. Burban} and \textit{O. Schiffmann} [Duke Math. J. 161, No. 7, 1171--1231 (2012; Zbl 1286.16029)].