The degree of the colored HOMFLY polynomial

Abstract

The colored HOMLFY polynomial is an important knot invariant depending on two variables $a$ and $q$. We give bounds on the degree in both $a$ and $q$ generalizing Morton's bounds \cite{Mo86} for the ordinary HOMFLY polynomial. Our bounds suggest that the degree detects certain incompressible surfaces in the knot complement and perhaps more generally features of the $SL(N)$ character varieties of the knot group. We formulate a precise conjecture along these lines generalizing the slope conjecture of Garoufalidis \cite{Ga11}. We prove our conjecture for all positive knots. Our technique is a reformulation of the MOY state sum \cite{MOY98} using $q$-analogues of Ehrhart polynomials. As a direct application we explicitly compute the $r$ coefficients of $r$-colored HOMFLY polynomial of any positive braid.