2014/12/31 by Roland van der Veen, van der Veen, Roland
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1501.00123
openalex publication_date 2014/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeMo86 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 \citeGa11. We prove our conjecture for all positive knots. Our technique is a reformulation of the MOY state sum \citeMOY98 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.