The degree of the colored HOMFLY polynomial

Understand

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}.