Constancy of the quadratic Jones-slope coefficient functions in the colored Jones degree quasipolynomials of a knot

For every knot K, the maximal and minimal degrees of the n-th colored Jones polynomial are, for all sufficiently large n, quadratic quasipolynomials in n, and their quadratic coefficient functions $s_2(n)$ and $s_2^*(n)$ are periodic with integral period. In every family of knots where these coefficients have ever been computed they are eventually constant, yet no theorem rules out non-constancy and no example exhibits it. Kalfagianni and Lee (Advances in Mathematics, 2023) singled the question out because the Jones diameter $jd_K$ — the invariant with which they characterized adequate knots ($jd_K = 2c(K)$) and determined crossing numbers of Whitehead doubles — has two competing definitions in the literature that agree when $s_2$ and $s_2^*$ are constant. The problem is to decide whether a knot with non-constant quadratic Jones-slope coefficient function exists, or to prove eventual constancy for all knots.

Background

Every knot K — an embedded circle in 3-dimensional space, considered up to continuous deformation — can be drawn as a diagram with finitely many transverse double points, and its crossing number $c(K)$ is the minimum number of such crossings over all diagrams of K. Computing $c(K)$ exactly is notoriously difficult: the sharp general lower bounds come from quantum invariants, and exact values are known only for knots admitting special diagram types. The Jones polynomial $V_K(q)$, and more generally the colored Jones polynomial $J_K(n)$ — the quantum invariant of K associated with the $n$-dimensional irreducible representation of the quantum group $sl_2$, with $J_K(2)$ the ordinary Jones polynomial — is a Laurent polynomial in q for each color n. Garoufalidis (2011) proved that the sequence $J_K(n)$ is q-holonomic, meaning it satisfies a linear recursion in n whose coefficients are rational functions of $q^n$ and q, and deduced that for n larger than some threshold $n_K$ the maximal and minimal degrees $d_+[J_K(n)]$ and $d_-[J_K(n)]$ are quadratic quasipolynomials — eventually periodic functions of n that agree with a quadratic polynomial on each residue class modulo some period. Concretely one can write

$$4d_+[J_K(n)] = s_2(n)n^2 + s_1(n)n + s_0(n), \;\; 4d_-[J_K(n)] = s_2^*(n)n^2 + s_1^*(n)n + s_0^*(n),$$

where each coefficient function is periodic in n with integral period. The values of the quadratic coefficient define the Jones slopes $js_K = \{s_2(n) : n > n_K\}$ and $js_K^* = \{s_2^*(n) : n > n_K\}$, finite sets of rational numbers, and the Jones diameter $jd_K = \max\{|s_2(n) - s_2^*(n)| : n > n_K\}$ measures the asymptotic spread of the degree. Garoufalidis's original definition of the Jones diameter instead maximizes $|s_2 - s_2^*|$ over all pairs of slope values attained possibly at different values of n; the two definitions agree when $s_2$ and $s_2^*$ are eventually constant. A diagram is adequate if the number of loops of the all-A and all-B Kauffman states — the diagrams obtained by resolving every crossing in one of the two possible ways — cannot be increased by changing any single resolution; knots admitting such diagrams, called adequate knots, include all alternating knots, and their adequate diagrams minimize crossings (Kauffman, Murasugi, Thistlethwaite). Kalfagianni and Lee (2023) proved $jd_K \le 2c(K)$ with equality if and only if K is adequate, and used this to determine the crossing numbers of untwisted Whitehead doubles of adequate knots and of their connected sums with adequate knots. The empirical substrate of this theory is striking: in every family where the degrees have been computed — alternating and adequate knots, torus knots, all knots up to 9 crossings, graph knots (cables and their iterated connected sums), Montesinos and pretzel families, cables, twisted generalized Whitehead doubles, and Mazur doubles — the quadratic coefficient functions $s_2, s_2^*$ are eventually constant, even though lower coefficients need not be (the linear coefficient has period 3 for the knots $8_{20}$, $9_{43}$, $9_{44}$, and the lower terms of the $(-2,3,7)$ pretzel knot's degree have period 4).

Problem Statement

Determine whether there exists a knot K for which the quadratic coefficient function $s_2(n)$ (or $s_2^*(n)$) in the eventually-quadratic quasipolynomial expansions of $4d_+[J_K(n)]$ and $4d_-[J_K(n)]$ — where $d_+[J_K(n)]$ and $d_-[J_K(n)]$ are the maximal and minimal degrees in q of the n-th colored Jones polynomial of K — is a non-constant periodic function of n, attaining at least two distinct rational values on residue classes of n beyond the stabilization threshold from which the quasipolynomial expansion holds. Equivalently: for every knot K, does each of the two sets of quadratic coefficient values attained by these expansions — the Jones slope sets $js_K = \{s_2(n) : n > n_K\}$ and $js_K^* = \{s_2^*(n) : n > n_K\}$ — consist of a single rational number? The question was singled out by Kalfagianni and Lee (Remark 2.6) because a non-constant example would be the first knot with more than one Jones slope, and could make the two literature definitions of the Jones diameter — and hence the adequacy criterion $jd_K = 2c(K)$ — genuinely definition-sensitive.

References

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