Freedman-He asymptotic crossing number conjecture: is the asymptotic crossing number of every knot equal to its crossing number?

Freedman and He introduced the asymptotic crossing number ac(K) of a knot K — the least over-crossing density achieved by pairs of high-degree satellite curves in a tubular neighborhood of K — as a topological invariant that lower-bounds the L^{3/2} energy of divergence-free vector fields leaving K invariant, and conjectured that ac(K) equals the ordinary crossing number c(K) for every knot, with an analogous statement for components of links. The conjecture implies the satellite crossing number question recorded as Kirby's Problem 1.68, which asks whether AC(K)=inf_d c(K_d)/d^2 equals c(K); the converse implication is not known. Despite partial results — the genus bound ac(K) ≥ 2g(K)−1, Lackenby's satellite bound c(K) ≥ c(k)/10^{13}, Ito's braid-index-dependent linear lower bound, and quadratic bounds for cables of adequate knots — the conjecture remains open in general.

Background

Physical origin. An ideal incompressible fluid or perfectly conducting plasma is modeled by a divergence-free vector field $W$ (one with $\nabla\cdot W=0$, so the field lines have no sources or sinks) transported by volume-preserving diffeomorphisms, i.e. smooth deformations of space that preserve the volume element. The helicity $H(W)=\int W\cdot(\nabla^{-1}W)\,dV$ (Moffatt's reformulation of Arnol'd's asymptotic Hopf invariant) measures the average pairwise linking of the field lines and is conserved under such transport (Moffatt 1990; Arnold-Khesin 1998). Moffatt's magnetic relaxation program asks how far a knotted or linked configuration of magnetic flux tubes can lower its energy while its topology is frozen in; helicity gives the first topological lower bound on that energy. Helicity is nevertheless a homological quantity, a function of pairwise linking numbers only, and is blind to knotting: the Whitehead link (a two-component link whose components have zero pairwise linking number yet cannot be pulled apart) is the standard example. Enciso, Peralta-Salas and Torres de Lizaur (2016) proved that every sufficiently regular integral invariant of volume-preserving transformations (a conserved quantity of the form $\int A(W)\,d\mathrm{vol}$) is a function of helicity alone, so knot-sensitive invariants of divergence-free fields must be of a different, non-integral kind.

Freedman and He's construction. In their 1991 Annals paper "Divergence-free fields: Energy and asymptotic crossing number", with the companion note "Links of tori and the energy of incompressible flows" (1991), they introduced such an invariant: the asymptotic crossing number of two divergence-free fields $W$ and $V$ is the unsigned Gauss-type double integral $c(W,V)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|\langle W(x),V(y),x-y\rangle|}{|x-y|^3}\,dx\,dy,$ where $\langle a,b,c\rangle$ denotes the scalar triple product; for a field supported on thin tubes it measures the average number of crossing points between the two families of field lines in a projection. Applying the sharp Hardy-Littlewood-Sobolev inequality (with Lieb's optimal constant), they proved the energy inequality $E_{3/2}(W)\ge(16/\pi)^{1/4}c(W,W)^{3/4}$, where $E_{3/2}$ is the $L^{3/2}$-energy of the field — the topological basis for Moffatt's proposed energy spectrum of knots and links. Since $c(W,W)$ is not itself invariant under volume-preserving diffeomorphisms, one defines $c_{\mathrm{top}}(W,W)=\inf_h c(h\circ W,h\circ W)$ over volume-preserving $h$, a genuine topological quantity that still lower-bounds the energy after relaxation.

Knot-theoretic input. Estimating $c_{\mathrm{top}}$ requires knot theory. When $W$ leaves invariant a solid torus $T$ of knot type $K$ carrying flux $\Phi$ (the situation of a knotted flux tube), an ergodic-theoretic averaging argument gives $c(W,W)\ge\Phi^2\operatorname{ac}(K)$, where $\operatorname{ac}(K)$ is the asymptotic crossing number of the knot $K$, defined by satellite links as follows. Let $N(K)$ be a closed tubular neighborhood of $K$, a solid torus, i.e. an embedded copy of $D^2\times S^1$. For integers $p,q\ge1$ let $c_{p,q}(K)$ be the minimum, over disjoint oriented closed curves $P,Q$ in the interior of $N(K)$ whose homological degrees in $H_1(N(K);\mathbb{Z})\con\mathbb{Z}$ are $p$ and $q$ (each curve wraps that many times around the core direction), and over all diagrams of the two-component link $(P,Q)$, of the number of crossings at which $P$ passes over $Q$. Then $\operatorname{ac}(K)=\inf_{p,q\ge1}c_{p,q}(K)/(pq)$, the least over-crossing density achievable by high-degree satellite links around $K$. For a link $L=(L_1,\dots,L_k)$ one defines $\operatorname{ac}(L_i;L)$ analogously, with the curve $Q$ allowed to run through the neighborhoods of the other components.

Known bounds and the conjecture. Freedman and He proved $\operatorname{ac}(K)\ge2g(K)-1$, where $g(K)$ is the Seifert genus (the minimal genus of a compact oriented surface spanning $K$, a Seifert surface), using the classical identification of the knot genus with the Thurston norm of the dual class (a theorem of Gabai; the Thurston norm of a homology class is the minimum, over embedded surfaces representing it, of the sum of the absolute Euler characteristics of their components); the link analogue is $\operatorname{ac}(L_i;L)\ge\chi_-(L_i;L)$, the negative Euler characteristic of a minimal spanning surface dual to the component. They conjectured the sharp statements (see the Problem Statement below). The satellite-knot reformulation is recorded as Problem 1.68 in Kirby's problem list: defining $AC(K)=\inf_d c(K_d)/d^2$ over satellite knots $K_d\subset N(K)$ of homological degree $d$ (i.e. winding $d$ times around the core $K$), is $AC(K)=c(K)$? The inequality $AC(K)\ge\operatorname{ac}(K)\ge2g(K)-1$ is known (recorded by Ito 2022), the upper bound $AC(K)\le c(K)$ is elementary, and Freedman-He's over-crossing form implies the satellite form; no implication in the other direction is known.

Consequences in conformal geometry. Freedman and He define the modulus of a solid torus $T$ as $m(T)=\inf_f\int_T|\nabla f|^3$ over Lipschitz maps $f:T\to\mathbb{R}/\mathbb{Z}$ of degree one, and prove the duality $m(T)^2m^*(T)=1$, where the dual modulus $m^*(T)$ is defined via divergence-free fields of unit flux in $T$, generalizing Gehring's classical ring-domain duality (Lohvansuu 2023), together with the bound $m(T)\le(\sqrt{\pi}/4)\operatorname{ac}(K)^{-3/2}$. Under the conjecture this yields that for each $m_0$ only finitely many knot types admit an embedded solid torus of modulus at least $m_0$, a compactness theorem for fat knotted tori not known unconditionally. The paper's adjacent question on the optimal conformal capacity of linked curves developed into the Gehring-Martin-Palka conjecture in higher-dimensional quasiconformal geometry, an active thread (Gehring-Martin-Palka 2017; Guerra-Prywes 2023; Lohvansuu 2023).

Status orientation. The knot-theoretic core of the conjecture, the exact quadratic behavior of the crossing numbers of high-degree satellites, has been attacked through the satellite crossing number literature (He 1998; Lackenby 2014; Ito 2022; Kalfagianni-McConkey 2024; Baker-Motegi 2024); the best general results are the genus bound, Lackenby's general satellite bound with a large multiplicative constant, a linear bound with braid-index-dependent coefficient, and quadratic bounds for cables of adequate knots.

Problem Statement

Definitions. Let $K\subset S^3$ be a knot and $N(K)$ a closed tubular neighborhood. For integers $p,q\ge1$, let $c_{p,q}(K)$ be the minimum number of crossings at which $P$ passes over $Q$, minimized over all diagrams of the two-component link $(P,Q)$ and over all pairs of disjoint oriented closed curves $P,Q\subset\operatorname{int}N(K)$ whose homological degrees in $H_1(N(K);\mathbb{Z})\con\mathbb{Z}$ are $p$ and $q$ (each curve wraps that many times around the core direction of the solid torus $N(K)$). The asymptotic crossing number of $K$ is $\operatorname{ac}(K)=\inf_{p,q\ge1}c_{p,q}(K)/(pq)$. For a link $L=(L_1,\dots,L_k)$, $\operatorname{ac}(L_i;L)$ is defined the same way with $P\subset N(L_i)$ of degree $p$ and $Q$ of degree $q$ in the union of the neighborhoods $N(L_j)$, $j\ne i$. Let $c(K)$ denote the minimal crossing number of $K$ (the least number of crossings in any diagram of $K$), and $\operatorname{cr}(L_i;L)$ the over-crossing number of $L_i$ over the other components, i.e. the minimum over all diagrams of $L$ of the number of crossings at which $L_i$ passes above some $L_j$, $j\ne i$.

Problem (Freedman-He asymptotic crossing number conjecture). Prove or refute: (1) for every knot $K$: $\operatorname{ac}(K)=c(K)$; (2) for every link $L$ and every component $L_i$: $\operatorname{ac}(L_i;L)=\operatorname{cr}(L_i;L)$.

Known without the conjecture: the easy direction $\operatorname{ac}(K)\le c(K)$ (draw degree-$p$ and degree-$q$ satellite curves as parallel strands along a minimal diagram of $K$), the genus bound $\operatorname{ac}(K)\ge2g(K)-1$ with $g(K)$ the Seifert genus (Freedman-He), the link bound $\operatorname{ac}(L_i;L)\ge\chi_-(L_i;L)$ (the negative Euler characteristic of a minimal spanning surface dual to the component), Ito's linear bound $\operatorname{ac}(K)\ge\frac{1}{f(b(K))}c(K)-b(K)$, where $b(K)$ is the braid index (the minimal number of strands of a braid whose closure is $K$) and $f$ is an explicit fast-growing function, and the quadratic bound $c(K_{p,q})>q^2c(K)$ for cables of adequate knots (the $(p,q)$-cable of $K$ is the satellite curve on $\partial N(K)$ with winding number $q$; adequate knots form a large class including the alternating knots, characterized by a combinatorial condition on their Kauffman states).

Equivalent and adjacent forms. The satellite-knot form, recorded as Kirby's Problem 1.68, asks whether $AC(K)=\inf_d c(K_d)/d^2$ equals $c(K)$, where $K_d$ ranges over satellite knots contained in $N(K)$ of homological degree $d$ (winding $d$ times around $K$); the inequality $AC(K)\ge\operatorname{ac}(K)$ is known, so statement (1) implies $AC(K)=c(K)$, while the upper bound $AC(K)\le c(K)$ is elementary, and no implication from the satellite form back to statement (1) is known. By Lackenby's analysis, the conjecture would also follow from sufficiently strong versions of the satellite crossing number conjecture (Kirby's Problem 1.67), for instance the winding-number quadratic bound $c(K)\ge\omega^2c(k)$ for every satellite $K$ of $k$ with winding number $\omega$ (He 1998).

Scope. Statement (1) is asserted for every knot and statement (2) for every link and every component, with no restriction to special classes; the conjecture remains open in general, and a single explicit counterexample to either (1) or (2) refutes it.

References

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