Additivity of the crossing number under connected sum of knots

The crossing number $c(K)$ of a knot $K$ is the minimum number of crossings over all diagrams of $K$. It is a long-standing open conjecture that $c$ is additive under connected sum: $c(K_1\#K_2)=c(K_1)+c(K_2)$ for all knots $K_1,K_2$. The upper bound $c(K_1\#K_2)\le c(K_1)+c(K_2)$ is trivial, but the best general lower bound known is linear with a small constant: $1/152$ (Lackenby, 2009), improved to a strict $1/16$ (Qiu–Wang, 2024). The open core is full additivity — equivalently, whether the optimal universal constant in the linear lower bound equals $1$ — together with any improvement of the constant beyond $1/16$.

Background

A knot is an embedded circle in 3-dimensional space, considered up to continuous deformation without cutting. A knot diagram is a generic projection to the plane recording over/under information at each double point; two diagrams of the same knot are related by the three Reidemeister moves. The crossing number $c(K)$ is the minimum number of crossings over all diagrams of $K$, the basic complexity measure of a knot. The connected sum $K_1\#K_2$ of two oriented knots, formed by cutting each open and splicing the ends together (well-defined for oriented knots by a theorem of Schubert), is a composite knot; knots not expressible as such a sum of two nontrivial knots are prime. Drawing minimal diagrams of $K_1$ and $K_2$ side by side and splicing gives $c(K_1\#K_2)\le c(K_1)+c(K_2)$, but no argument is known that even gives $c(K_1\#K_2)\ge c(K_1)$ in general. Additivity is known when both factors are alternating (Kauffman 1987 and Murasugi 1987, via the span of the Jones polynomial — a polynomial knot invariant that is multiplicative under connected sum and whose span (the difference between its highest and lowest exponents) bounds $4\,c(K)$ with equality for alternating knots), when both are torus knots (knots lying on the surface of a standard unknotted torus in 3-space), and for certain link sums (Diao 2004). The breakthrough for general knots is Lackenby (2009): using normal surface theory (surfaces in a triangulated 3-manifold placed in a controlled normal form) together with generalized Heegaard splittings (decompositions of the 3-manifold into simple handlebody pieces) and parallelity bundles (product neighborhoods of surfaces, in which normal surfaces decompose into controlled blocks) of the knot complement, he proved the universal linear bound $(c(K_1)+\cdots+c(K_n))/152\le c(K_1\#\cdots\#K_n)$ for any number of summands, and announced an analogous satellite-knot bound (a satellite knot $K$ lies essentially inside a solid-torus neighborhood of a nontrivial companion $L$). The satellite announcement was later carried out by Lackenby (2014) with the explicit constant $10^{13}$, i.e. $c(K)\ge c(L)/10^{13}$, while the sharper conjecture $c(K)\ge c(L)$ remains open. The composite constant was improved by Qiu–Wang (2024) from $1/152$ to a strict $1/16$ of the sum of the factor crossing numbers. Determining the optimal universal constant — in particular whether it equals $1$ — is the classical additivity conjecture, open since the 1950s and recorded on standard low-dimensional topology problem lists.

Problem Statement

Prove or disprove the crossing number additivity conjecture for composite knots: for all nontrivial oriented knots $K_1$ and $K_2$, $c(K_1\#K_2)=c(K_1)+c(K_2),$ with the analogous equality for connected sums of any finite number of factors. Equivalently, determine the optimal universal constant $\alpha$ such that $c(K_1\#K_2)\ge\alpha\,(c(K_1)+c(K_2))$ holds for all $K_1,K_2$: currently $1/16\le\alpha\le 1$, with $\alpha=1$ exactly the additivity conjecture.

References

1. Marc Lackenby, The crossing number of composite knots, Journal of Topology 2 (2009) 747–768, DOI 10.1112/jtopol/jtp028, https://doi.org/10.1112/jtopol/jtp028 (arXiv:0805.4706, https://arxiv.org/abs/0805.4706)
2. Marc Lackenby, The crossing number of satellite knots, Algebraic & Geometric Topology 14 (2014) 2379–2409, DOI 10.2140/agt.2014.14.2379, https://doi.org/10.2140/agt.2014.14.2379 (arXiv:1106.3095, https://arxiv.org/abs/1106.3095)
3. Ruifeng Qiu and Chao Wang, A lower bound of the crossing number of composite knots, arXiv:2407.10467, https://arxiv.org/abs/2407.10467
4. 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
5. 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
6. 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
7. Kenneth L. Baker and Kimihiko Motegi, The stable crossing number of a twist family of knots and the satellite crossing number conjecture, arXiv:2404.05308, https://arxiv.org/abs/2404.05308