  We give a generators and relations presentation of the HOMFLYPT skein algebra
$H$ of the torus $T^2$, and we give an explicit description of the module
corresponding to the solid torus. Using this presentation, we show that $H$ is
isomorphic to the $t=q$ specialization of the elliptic Hall algebra of Burban
and Schiffmann [BS12].
  As an application, for an iterated cable $K$ of the unknot, we use the
elliptic Hall algebra to construct a 3-variable polynomial that specializes to
the $\lambda$-colored Homflypt polynomial of $K$. We show that this polynomial
also specializes to one constructed by Cherednik and Danilenko [CD14] using the
$\mathfrak{gl}_N$ double affine Hecke algebra. This proves one of the
Connection Conjectures in [CD14].
