Monomial web basis for the SL(N) skein algebra of the twice punctured sphere

Abstract

We give a new proof of a slightly modified version of a result of Queffelec--Rose, by constructing a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere for any non-zero complex number $q$, excluding finitely many roots of unity of small order. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon--Wong in the case $n=2$. Two consequences of our method are that the quantum trace map and the so-called splitting map embed the polynomial algebra into the Fock--Goncharov quantum higher Teichm\"uller space and the L\^{e}--Sikora stated skein algebra, respectively, of the annulus. We end by discussing the relationship with Fock--Goncharov duality.