The Coarse Volume Conjecture: does a function of the colored Jones coefficients coarsely determine the hyperbolic volume of every hyperbolic knot?

For a knot K in the 3-sphere, the colored Jones polynomials form a sequence of quantum invariants J_K^n(t), n = 1, 2, 3, ..., while a hyperbolic knot complement S^3 \ K carries a canonical hyperbolic metric with a well-defined volume vol(S^3 \ K). Futer, Kalfagianni, and Purcell posed the Coarse Volume Conjecture (Question 10.13 of their monograph 'Guts of surfaces and the colored Jones polynomial'): does there exist a function B(K) of the coefficients of the colored Jones polynomials of K such that, over the class of all hyperbolic knots, B(K) is coarsely related to the hyperbolic volume — i.e. there are universal constants C1 >= 1 and C2 >= 0 with C1^(-1) B(K) - C2 <= vol(S^3 \ K) <= C1 B(K) + C2 for every hyperbolic knot K? Two-sided linear bounds of exactly this shape are known only for diagrammatically restricted families (alternating/adequate knots, highly twisted positive braid closures and Montesinos links, semi-adequate closed braids and plats, highly twisted near-alternating knots, and generalized alternating links on surfaces), and it is known that no function of only the second and penultimate Jones coefficients can coarsely predict volume for all hyperbolic knots. The conjecture for the unrestricted class of hyperbolic knots remains open.

Background

A knot K is a smoothly embedded circle in the 3-sphere S^3. To each knot, quantum topology associates the Jones polynomial J_K(t) (a Laurent polynomial introduced by Vaughan Jones in 1985) and its higher-color generalizations, the colored Jones polynomials J_K^n(t): a sequence of Laurent polynomials in a variable t, one for each positive integer n, with J_K^2(t) essentially the ordinary Jones polynomial. These are combinatorial 'quantum' invariants, computable from any diagram of the knot via Kauffman-bracket skein relations.

Independently, a knot whose complement S^3 \ K admits a complete hyperbolic metric of finite volume is called a hyperbolic knot; by Thurston's geometrization theorem every knot that is neither a torus knot nor a satellite knot is hyperbolic, and by Mostow–Prasad rigidity its hyperbolic volume vol(S^3 \ K) is a topological invariant of the knot. The hyperbolic volume of a hyperbolic knot complement is bounded below by the volume of the figure-eight knot complement (about 2.03, the unique minimum among hyperbolic knot complements, by the Cao–Meyerhoff theorem) and is unbounded above — for instance, for alternating and adequate knots it grows linearly with the number of twist regions of a diagram — so understanding which quantum data 'sees' this geometry, even up to coarse multiplicative bounds, is a central problem.

The Kashaev–Murakami–Murakami volume conjecture predicts that the asymptotic growth of J_K^n evaluated at an n-th root of unity yields vol(S^3 \ K) in the limit; it remains open. In the 2000s, Dasbach and Lin discovered a coarser, unconditional phenomenon for alternating knots: writing beta'_K and beta''_K for the second and penultimate coefficients of the Jones polynomial (equivalently, of J_K^n for alternating knots, where the extreme coefficients stabilize in n), the sum |beta'_K| + |beta''_K| bounds the hyperbolic volume from both sides up to multiplicative constants. Futer, Kalfagianni, and Purcell extended such two-sided 'volume-ish' bounds to all adequate knots (knots admitting a diagram whose all-A and all-B Kauffman states have diagrammatic properties making the associated 'state surfaces' essential in the knot complement), to closures of positive braids with at least three crossings per twist region (a twist region being a maximal string of bigons between two crossings), and to Montesinos links. A knot is adequate when both of its checkerboard state graphs survive a certain edge reduction without loops.

Two facts delimit what is known. First, all proven bounds route coefficient data through the twist number of a diagram (the number of twist regions), and both steps of that route are known to fail outside special families of diagrams. Second, Futer–Kalfagianni–Purcell proved (via families of adequate Montesinos knots and closed 3-braids) that no function of the two Jones coefficients beta'_K, beta''_K alone can coarsely predict the volume of all hyperbolic knots — so any coefficient-level function B(K) must draw on more of the colored Jones coefficient data. A theorem of Garoufalidis and Lê (the colored Jones function is q-holonomic) implies that for a fixed knot the entire sequence of polynomials J_K^n(t) is determined by finitely many of them, so their coefficients satisfy linear recursions; this makes the coefficient data a finite, well-defined input, although the head and tail coefficients are known to stabilize only for special classes, and their eventual periodicity for arbitrary knots is itself an open companion question (Question 10.14 of the same monograph).

In Section 10.4 of 'Guts of surfaces and the colored Jones polynomial', Futer, Kalfagianni, and Purcell formalize the paradigm. Functions f, g : Z -> R+ on a set Z are called coarsely related if there exist universal constants C1 >= 1 and C2 >= 0 with C1^(-1) f(x) - C2 <= g(x) <= C1 f(x) + C2 for all x in Z (the notion used throughout coarse geometry, e.g. for quasi-isometric embeddings). Taking Z to be the infinite set of hyperbolic knots, they pose Question 10.13, the Coarse Volume Conjecture.

Problem Statement

Coarse Volume Conjecture (Futer–Kalfagianni–Purcell, Question 10.13). Does there exist a function $B(K)$, depending only on the coefficients of the colored Jones polynomials $J_K^n(t)$ ($n = 1, 2, 3, \dots$) of a knot $K$, such that $B(K)$ is coarsely related to hyperbolic volume over the class of all hyperbolic knots? That is, do there exist universal constants $C_1 \ge 1$ and $C_2 \ge 0$, independent of $K$, such that for every hyperbolic knot $K$ in the 3-sphere,

$C_1^{-1}\, B(K) - C_2 \;\le\; \operatorname{vol}(S^3 \setminus K) \;\le\; C_1\, B(K) + C_2,$

where $\operatorname{vol}(S^3 \setminus K)$ is the hyperbolic volume of the knot complement? The conjecture places no restriction on which coefficients of the colored Jones sequence $B$ may use (but $B$ may not use other data, such as a choice of diagram); it is known that $B$ cannot be a function of the second and penultimate Jones coefficients alone, and the answer is affirmative on restricted families (alternating and adequate knots, semi-adequate closed braids and plat closures, highly twisted near-alternating knots, generalized alternating links on surfaces). The conjecture concerns the entire class of hyperbolic knots: either such a function $B$ exists, with both inequalities holding for every hyperbolic knot, or no function of the colored Jones coefficients is coarsely related to hyperbolic volume on that class.

References

1. David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Guts of surfaces and the colored Jones polynomial, Lecture Notes in Mathematics 2069, Springer, Berlin, 2013, ISBN 978-3-642-33301-9; DOI 10.1007/978-3-642-33302-6; arXiv:1108.3370, https://arxiv.org/abs/1108.3370 (the question is Question 10.13 in Section 10.4)
2. Oliver T. Dasbach, Xiao-Song Lin, On the Head and the Tail of the Colored Jones Polynomial, Compositio Mathematica 142 (2006), no. 5, 1332–1342; arXiv:math/0604230, https://arxiv.org/abs/math/0604230
3. David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Dehn filling, volume, and the Jones polynomial, Journal of Differential Geometry 78 (2008), no. 3, 429–464; arXiv:math/0612138, https://arxiv.org/abs/math/0612138
4. David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Symmetric links and Conway sums: volume and Jones polynomial, Mathematical Research Letters 16 (2009), no. 2, 233–253; arXiv:0804.1542, https://arxiv.org/abs/0804.1542
5. David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Cusp areas of Farey manifolds and applications to knot theory, International Mathematics Research Notices 2010 (2010), no. 23, 4434–4497; arXiv:0808.2716, https://arxiv.org/abs/0808.2716
6. David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Jones polynomials, volume, and essential knot surfaces: a survey, Banach Center Publications 100 (2014), 51–77 (Proceedings of Knots in Poland III); arXiv:1110.6388, https://arxiv.org/abs/1110.6388
7. Adam Giambrone, Combinatorics of link diagrams and volume, Journal of Knot Theory and Its Ramifications 24 (2015), no. 1, 1550001; DOI 10.1142/S0218216715500017; arXiv:1310.8414, https://arxiv.org/abs/1310.8414
8. Adam Giambrone, Semi-Adequate Closed Braids and Volume, Topology and its Applications 198 (2016), 1–21; DOI 10.1016/j.topol.2015.10.009; arXiv:1406.7337, https://arxiv.org/abs/1406.7337
9. Christine Ruey Shan Lee, Jones slopes and coarse volume of near-alternating links, arXiv:1708.04900, https://arxiv.org/abs/1708.04900
10. Brandon Bavier, Efstratia Kalfagianni, Guts, volume and skein modules of 3-manifolds, Fundamenta Mathematicae 256 (2022), no. 2, 195–220; DOI 10.4064/fm996-1-2021; arXiv:2010.06559, https://arxiv.org/abs/2010.06559
11. Hitoshi Murakami, Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Mathematica 186 (2001), no. 1, 85–104; DOI 10.1007/BF02392716; arXiv:math/9905075, https://arxiv.org/abs/math/9905075
12. Stavros Garoufalidis, Thang T. Q. Lê, The colored Jones function is q-holonomic, Geometry & Topology 9 (2005), 1253–1293; arXiv:math/0309214, https://arxiv.org/abs/math/0309214
13. Christine Ruey Shan Lee, Stability properties of the colored Jones polynomial, Journal of Knot Theory and Its Ramifications 28 (2019), no. 8, 1950050; DOI 10.1142/S0218216519500500; arXiv:1409.4457, https://arxiv.org/abs/1409.4457
14. Daniel Ruberman, Mutation and volumes of knots in S^3, Inventiones Mathematicae 90 (1987), no. 1, 189–215; DOI 10.1007/BF01389038, https://doi.org/10.1007/BF01389038