2005/08/31 by Stavros Garoufalidis, Thang T. Q. Lê, Thang T. Q. Le · 65 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Colored #Combinatorics #Conjecture #Geometric and Algebraic Topology #Infinity #Jones polynomial #Knot (papermaking) #Knot theory #Mathematical analysis #Mathematics #Polynomial #Pure mathematics #Upper and lower bounds #math.GT #math.QA #msc:57M25 #msc:57N10 #semigroups and automata theory
paper · pdf · doi:10.2140/gt.2011.15.2135
published in Geometry & Topology 15(4), 2135-2180 (Mathematical Sciences Publishers) · 31 pages, 13 figures
arxiv created 2011/08/31 · openalex publication_date 2011/10/28 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose n-th term is the n-th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the n-th colored Jones polynomial at e =n , when is a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1=n when is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the n-th colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol, Dunfield, Storm and W Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when is near 2 i . Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.