On the head and the tail of the colored Jones polynomial
By Oliver T. Dasbach, Xiaoxia Lin. Compositio Mathematica, 2006
85 citations recorded by the source corpus.
The colored Jones polynomial is a function $J_K:{\mathbb{N}}\longrightarrow{\mathbb{Z}}[t,t^{-1}]$ associated with a knot $K$ in 3-space. We will show that for an alternating knot $K$ the absolute values of the first and the last three leading coefficients of $J_K(n)$ are independent of $n$ when $n$ is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a volume-ish theorem for the colored Jones polynomial.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0ae9ab38250d6b342c09136087387e5fc64ad851
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial (2011, background): “atural number N. For each N the colored Jones polynomial J N;K(q) is normalized to be 1 for Kthe unknot. The ordinary Jones polynomial is J 2;K(q). It was shown by Xiao-Song Lin and the second author [DL06] that for an alternating link Kthe three leading coecients of the colored Jones polynomial J N”
- The head and tail conjecture for alternating knots (2011, background): “In [2] and [3] Oliver Dasbach and Xiao-Song Lin showed that, up to sign, the first two coefficients and the last two coefficients of JN,K(q) do not depend on N for alternating knots.”