The satellite crossing number conjecture

The satellite crossing number conjecture (Problem 1.67 in Kirby's 1995 problem list) asserts that a satellite knot can never be simpler than its companion: if $K$ is a non-trivial satellite knot with companion $C$, then $c(K) \ge c(C)$, where $c$ denotes crossing number. Standard strengthenings predict $c(K) \ge \omega^2 c(C)$ for patterns of winding number $\omega \ge 1$ and $c(K) \ge \eta^2 c(C)$ for wrapping number $\eta$. Since a connected sum is a winding-number-1 satellite, the conjecture implies $c(K_1 \# K_2) \ge \max(c(K_1), c(K_2))$, the inequality half of the classical additivity conjecture. The source paper is Kalfagianni–McConkey's "Crossing numbers of cable knots" (Bull. London Math. Soc. 2024), which proved $c(K_{p,q}) \ge q^2 c(K)+1$ for cables of adequate knots and determined the exact crossing numbers of 2-cables of adequate knots while stating the general satellite conjecture as open. As of August 2026 the conjecture remains open in all generality. The strongest bounds valid for every satellite are Lackenby's $c(K) \ge c(C)/10^{13}$ and, for composites, Qiu–Wang's $c(K_1 \# \cdots \# K_n) > \frac{1}{16}\sum_i c(K_i)$; recent partial results cover cables of adequate knots, Mazur doubles, highly twisted satellites (Baker–Motegi), and large-winding satellites with bounded companion braid index (Ito). A very recent unrefereed preprint (Kawauchi, August 2026) claims full additivity via 4D topology, but the general satellite conjecture remains open.

Background

The crossing number $c(K)$ of a knot $K$ — a knotted closed curve in the 3-sphere $S^3$ — is the minimum number of crossings over all planar diagrams of $K$. It is the oldest complexity measure of a knot and the basis of knot tabulation, yet it is notoriously hard to control: there is no practical algorithm computing it for an arbitrary knot, and its exact value is known only for special classes, including alternating knots and more generally adequate knots, torus knots, certain Montesinos knots, and a handful of satellite families.

A satellite knot is produced by the most basic "knot inside a knot" construction. A pattern is a knot $P$ inside an unknotted solid torus $V \subset S^3$ that is not contained in a 3-ball of $V$ and is not a core curve of $V$. Embedding $V$ as a tubular neighbourhood $N(C)$ of a non-trivial companion knot $C$ yields the satellite $K = P(C)$. Two integer parameters of the pattern recur throughout: the winding number $\omega$, the algebraic intersection number of $P$ with a meridian disk of $V$, and the wrapping number $\eta \ge 1$, the minimal unsigned such intersection number (a pattern not contained in a 3-ball has $\eta \ge 1$; for the connected-sum pattern below, $\eta = 1$). Standard examples: the $(p,q)$-cable of $K$ is the simple closed curve on $\partial N(K)$ wrapping $p$ times around the meridian and $q$ times around the canonical longitude (winding number $q$), and the connected sum $K_1 \# K_2$ is the satellite of $K_1$ with pattern $K_2$, of winding number 1.

The conjecture asserts that this construction can never simplify a knot below its companion:

$c(K) \ge c(C) \quad \text{for every non-trivial satellite } K \text{ with companion } C.$

It is Problem 1.67 in Kirby's 1995 problem list and is usually stated together with two strengthenings: $c(K) \ge \omega^2 c(C)$ for winding number $\omega \ge 1$ patterns, and $c(K) \ge \eta^2 c(C)$ for wrapping number $\eta$ patterns (the winding-number strengthening would also settle Freedman and He's conjecture that the asymptotic crossing number of a knot — the infimum of $c(K_w)/w^2$ over winding-number-$w$ satellites — equals its crossing number, Kirby Problem 1.68). The heuristic is visible in diagrams: a satellite diagram contains a copy of a companion diagram inside the region bounded by the image of $\partial V$, and a winding-$\omega$ pattern crosses each meridian disk $\omega$ times, so one expects at least on the order of $\omega^2 c(C)$ crossings; for cables of adequate knots this count is exactly realized, as $c(K_{p,q}) \ge q^2 c(K)+1$ (Kalfagianni–McConkey). Because a connected sum is a winding-number-1 satellite, the conjecture contains the inequality $c(K_1 \# K_2) \ge c(K_1)$ — the lower-bound half of the classical additivity conjecture, usually attributed to Fox, that $c(K_1 \# K_2) = c(K_1) + c(K_2)$. That equality is a separate open problem not settled by the satellite conjecture.

Three method families account for all known partial results, and each explains a facet of the difficulty. (1) Polynomial-span methods. The Kauffman bracket polynomial — the unnormalised precursor of the Jones polynomial, a state sum obtained by resolving each crossing in one of two ways — has degree span at most $4c(K)$ for every knot, with equality whenever $K$ is adequate, meaning $K$ has a diagram whose all-A and all-B state graphs (vertices = state circles of the all-A / all-B resolutions, edges = crossings) contain no one-edged loops. This is the Kauffman–Murasugi–Thistlethwaite solution of Tait's conjectures, extended to adequate links by Lickorish and Thistlethwaite, and it makes crossing numbers exactly computable precisely where the equality holds. Kalfagianni and Lee's Jones-diameter inequality $dj_K \le 2c(K)$ — where $dj_K$ is the limiting, when the limit exists, normalized degree diameter of the $n$-th colored Jones polynomial, the quantum invariant coloring the knot by the $n$-dimensional irreducible representation of $sl_2$, with equality again characterizing adequate knots — extends such exactness to certain non-adequate satellites and is the engine behind the source paper's cable results and behind Baker–Motegi–Takata's determination of the crossing numbers of Mazur doubles of adequate knots up to one crossing. (2) Geometric methods. Lackenby's normal-surface and handle-structure arguments give linear bounds valid for all satellites but with astronomical constants. (3) Braid-theoretic methods. Ito's quantitative Birman–Menasco finiteness theorem bounds crossing numbers in terms of genus and braid index, yielding the conjecture when the winding number and the companion's crossing number dominate the companion's braid index.

Despite all this, the general conjecture — and even special cases such as the composite inequality $c(K_1 \# K_2) \ge c(K_1)$ or the "cabling crossing number conjecture" $c(K_{p,q}) \ge c(K)$ for cables of arbitrary knots (Ito) — remains open.

Problem Statement

Prove or refute the Satellite Crossing Number Conjecture: for every knot $K$ that is a non-trivial satellite with companion $C$ — i.e. $K$ lies in a tubular neighbourhood $N(C)$ of the non-trivial knot $C$, is not contained in a 3-ball in $N(C)$, and is not a core curve of $N(C)$ — the crossing numbers satisfy

$c(K) \ge c(C).$

A proof must hold for all companions and all satellite patterns, with no restriction to adequate or alternating knots, cables, positive winding number, hyperbolic companions, or any other special class. The standard strengthenings — $c(K) \ge \omega^2 c(C)$ when the pattern has winding number $\omega \ge 1$ (the algebraic intersection number of the pattern with a meridian disk of the solid torus containing it), and $c(K) \ge \eta^2 c(C)$ when the pattern has wrapping number $\eta$ (the minimal number of unsigned intersections with such a disk) — are part of the conjecture package, and a resolution is expected to establish or explicitly delimit them, but the completion boundary of this problem is the core inequality $c(K) \ge c(C)$ (Kirby's Problem 1.67). A refutation consists of an explicit non-trivial satellite $K$ with companion $C$ together with a certified crossing number of $K$ strictly smaller than a certified crossing number of $C$.

Scope note: since a connected sum $K_1 \# K_2$ is a satellite of $K_1$ with winding number 1, the conjecture implies $c(K_1 \# K_2) \ge \max(c(K_1), c(K_2))$ for all knots; it does not by itself imply the additivity conjecture $c(K_1 \# K_2) = c(K_1)+c(K_2)$, which is a separate question and not part of this problem.

References

1. Efstratia Kalfagianni and Rob McConkey, 'Crossing numbers of cable knots', Bulletin of the London Mathematical Society 56 (2024) 3400–3411, DOI 10.1112/blms.13140, arXiv:2309.03814, https://arxiv.org/abs/2309.03814
2. Robion Kirby, 'Problems in low-dimensional topology', in W. H. Kazez (ed.), Geometric topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics vol. 2, part 2, American Mathematical Society, Providence, RI, and International Press, Cambridge, MA, 1997, pp. 35–473, ISBN 0-8218-0654-8, Zbl 0888.57014, https://zbmath.org/0888.57014
3. Marc Lackenby, 'The crossing number of satellite knots', Algebraic & Geometric Topology 14 (2014) 2379–2409, DOI 10.2140/agt.2014.14.2379, arXiv:1106.3095, https://arxiv.org/abs/1106.3095
4. Marc Lackenby, 'The crossing number of composite knots', Journal of Topology 2 (2009) 747–768, DOI 10.1112/jtopol/jtp028, arXiv:0805.4706, https://arxiv.org/abs/0805.4706
5. Efstratia Kalfagianni and Christine Ruey Shan Lee, 'Jones diameter and crossing number of knots', Advances in Mathematics 417 (2023) article 108937, DOI 10.1016/j.aim.2023.108937, arXiv:2108.12391, https://arxiv.org/abs/2108.12391
6. Tetsuya Ito, 'A quantitative Birman–Menasco finiteness theorem and its application to crossing number', Journal of Topology 15 (2022) 1794–1806, DOI 10.1112/topo.12259, arXiv:2010.12150, https://arxiv.org/abs/2010.12150
7. Zheng-Xu He, 'On the crossing number of high degree satellites of hyperbolic knots', Mathematical Research Letters 5 (1998) 235–246, DOI 10.4310/MRL.1998.v5.n2.a10, https://doi.org/10.4310/MRL.1998.v5.n2.a10
8. Michael H. Freedman and Zheng-Xu He, 'Divergence-free fields: energy and asymptotic crossing number', Annals of Mathematics (2) 134 (1991) 189–229, DOI 10.2307/2944336, https://doi.org/10.2307/2944336
9. Yuanan Diao, 'The additivity of crossing numbers', Journal of Knot Theory and Its Ramifications 13 (2004) 857–866, DOI 10.1142/S0218216504003524, https://doi.org/10.1142/S0218216504003524
10. Kunio Murasugi, 'Jones polynomials and classical conjectures in knot theory', Topology 26 (1987) 187–194, DOI 10.1016/0040-9383(87)90058-9, https://doi.org/10.1016/0040-9383(87)90058-9
11. Louis H. Kauffman, 'State models and the Jones polynomial', Topology 26 (1987) 395–407, DOI 10.1016/0040-9383(87)90009-7, https://doi.org/10.1016/0040-9383(87)90009-7
12. Morwen B. Thistlethwaite, 'Kauffman's polynomial and alternating links', Topology 27 (1988) 311–318, DOI 10.1016/0040-9383(88)90012-2, https://doi.org/10.1016/0040-9383(88)90012-2
13. W. B. R. Lickorish and M. B. Thistlethwaite, 'Some links with non-trivial polynomials and their crossing-numbers', Commentarii Mathematici Helvetici 63 (1988) 527–539, DOI 10.1007/BF02566777, https://doi.org/10.1007/BF02566777
14. A. Stoimenow, 'On the satellite crossing number conjecture', Journal of Topology and Analysis 3 (2011) 109–143, DOI 10.1142/S1793525311000568, https://doi.org/10.1142/S1793525311000568
15. Kenneth L. Baker, Kimihiko Motegi, and Toshie Takata, 'The Strong Slope Conjecture and crossing numbers for Mazur doubles of knots', in M. Morishita, H. Nakamura, and J. Ueki (eds.), Low Dimensional Topology and Number Theory, Springer Proceedings in Mathematics & Statistics vol. 456, Springer, 2025, pp. 1–61, DOI 10.1007/978-981-97-3778-9_1, arXiv:2204.05725, https://doi.org/10.1007/978-981-97-3778-9_1
16. Kenneth L. Baker and Kimihiko Motegi, 'The stable crossing number of a twist family of knots and the satellite crossing number conjecture', Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (published online December 2024), DOI 10.2422/2036-2145.202404_011, arXiv:2404.05308, https://arxiv.org/abs/2404.05308
17. Ruifeng Qiu and Chao Wang, 'A lower bound of the crossing number of composite knots', preprint (2024), arXiv:2407.10467, https://arxiv.org/abs/2407.10467
18. Akio Kawauchi, 'The crossing numbers of knots and links via 4D topology', preprint (August 2026), arXiv:2608.13789, https://arxiv.org/abs/2608.13789
19. Adrián Jiménez Pascual, 'On wrapping number, adequacy and the crossing number of satellite knots', unpublished preprint (2017), arXiv:1712.05635, https://arxiv.org/abs/1712.05635