Additivity of the crossing number of knots under connected sum (Tait's problem; Kirby Problem 1.65)

The crossing number $c(K)$ of a knot $K$ is the minimum number of crossings over all diagrams of $K$. It is elementary that $c(K_1\#K_2)\le c(K_1)+c(K_2)$ for the connected sum $K_1\#K_2$, but the reverse inequality — additivity of the crossing number, a question going back to P. G. Tait in the 1870s and recorded as Problem 1.65 in Kirby's problem list — remains unproven for general knots. The best published universal lower bound was improved from $c(K_1\#\cdots\#K_n)\ge(\sum_i c(K_i))/152$ (Lackenby, 2009) to $> (\sum_i c(K_i))/16$ (Qiu–Wang, 2024, revised 31 July 2026), with the latter still stating the exact conjecture as unsolved. A preprint posted 13 August 2026 by A. Kawauchi (arXiv:2608.13789) claims a full proof of additivity for all knots and links via an equality between crossing numbers and chord indices of spun ribbon torus-links in the 4-sphere; that claim is unrefereed, uncited, and unverified by the community, so the problem is treated here as open. (Additivity is Kirby Problem 1.65; Problem 1.68 concerns the asymptotic crossing number $AC(K)$.)

Background

A knot $K$ is a smooth embedding of a circle in the 3-sphere $S^3$, taken up to ambient isotopy (a continuous deformation of the surrounding space). A knot diagram is a generic projection of $K$ onto a 2-sphere or plane, recording at each double point (crossing) which strand passes over; the crossing number $c(K)$ is the minimum number of crossings over all diagrams of $K$. The connected sum $K_1\#K_2$ of two oriented knots is formed by deleting a small arc from each and gluing the resulting tangle boundaries orientation-preservingly; for oriented knots in $S^3$ the result is well defined up to isotopy (Schubert's theorem). Combining minimal diagrams of $K_1$ and $K_2$ always gives a diagram of $K_1\#K_2$, so $c(K_1\#K_2)\le c(K_1)+c(K_2)$ is immediate. The conjecture asserts equality. The question is old: P. G. Tait assumed such additivity-like minimality behavior in his 19th-century knot tables, and R. Kirby recorded it as Problem 1.65 of his 1995 problem list in low-dimensional topology ('Is the crossing number $c(K)$ of a knot $K$ additive with respect to connected sum, that is, is the equality $c(K_1\#K_2)=c(K_1)+c(K_2)$ true?'). Additivity is known for large classes: Murasugi (1987), Kauffman (1987) and Thistlethwaite (1987) independently proved the Tait conjecture that reduced alternating diagrams (alternating diagrams without nugatory crossings, i.e. crossings removable by a simple untwisting move) are minimal, using the Jones polynomial (a quantum invariant of diagrams introduced in 1984 whose span — the difference between its highest and lowest exponents — gives a lower bound on the crossing number and is additive under connected sum); this yields $c(K_1\#K_2)=c(K_1)+c(K_2)$ whenever both summands are alternating, and the same argument extends to adequate knots (knots admitting a diagram whose Kauffman-bracket state circles attain the extremal size needed for the span argument, a class containing all alternating knots) and to Diao's zero-deficiency knots, defined by equality in a Jones-polynomial deficiency inequality and including all torus knots. For general knots, only non-sharp lower bounds are known: Lackenby (2009), via normal-surface techniques (the study of surfaces in a triangulated 3-manifold meeting each tetrahedron in standard triangles and squares) on Heegaard splittings (decompositions of a 3-manifold into two handlebodies) of the composite complement, proved $c(K_1\#\cdots\#K_n)\ge(c(K_1)+\cdots+c(K_n))/152$; Ito (2022) proved a lower bound depending on the braid indices of the summands (the braid index is the least number of strands of a braid whose closure is the knot), sharper than Lackenby's when the braid indices are small; and Qiu–Wang (2024, revised July 2026), refining Lackenby's strategy with new handle-structure and normal-surface techniques, proved $c(K_1\#\cdots\#K_n)>(c(K_1)+\cdots+c(K_n))/16$. The conjecture is closely related to the stronger satellite conjecture (Kirby Problem 1.67: the crossing number of a nontrivial satellite knot exceeds that of its companion — a satellite knot being one embedded nontrivially inside a tubular neighborhood of another knot, its companion): since $K_1\#K_2$ is a satellite of $K_1$, the satellite conjecture restricted to this case would give the weaker inequality $c(K_1\#K_2)\ge c(K_1)$, which is itself open. Note on the Kirby numbering: in Kirby's list, Problem 1.68 is the asymptotic crossing number conjecture $AC(K)=c(K)$ with $AC(K)=\inf_d c(K_d)/d^2$ over satellites of homological degree $d$; the additivity problem is Problem 1.65 (author-hosted problem list: https://math.berkeley.edu/~kirby/problems.ps.gz).

Problem Statement

Prove or disprove: for any two oriented knots $K_1$ and $K_2$ in the 3-sphere, the crossing number is additive under connected sum, i.e. $c(K_1\#K_2)=c(K_1)+c(K_2)$. Since $c(K_1\#K_2)\le c(K_1)+c(K_2)$ holds by combining diagrams, the open content is exactly the lower bound $c(K_1\#K_2)\ge c(K_1)+c(K_2)$ for all oriented knots (equivalently, for every nontrivial composite knot $K=K_1\#\cdots\#K_n$, the inequality $c(K)\ge c(K_1)+\cdots+c(K_n)$). A disproof consists of an explicit pair of knots with $c(K_1\#K_2)<c(K_1)+c(K_2)$, together with the crossing numbers of the three knots. This is Problem 1.65 in Kirby's problem list (often traced back to Tait); the natural extension to connected sums of nontrivial links, where the connected sum depends on the choice of components, may be settled by the same answer but is not part of the classical knot conjecture.

References

1. R. Kirby, 'Problems in low-dimensional topology', in Geometric topology (Athens, GA, 1993), AMS/IP Studies in Advanced Mathematics vol. 2.2, American Mathematical Society, 1997, pp. 35–473 (Problem 1.65). Author-hosted source: https://math.berkeley.edu/~kirby/problems.ps.gz
2. M. Lackenby, 'The crossing number of composite knots', Journal of Topology 2 (2009), no. 4, 747–768. DOI: 10.1112/jtopol/jtp028, https://doi.org/10.1112/jtopol/jtp028 (preprint arXiv:0805.4706, https://arxiv.org/abs/0805.4706)
3. R. Qiu and C. Wang, 'A lower bound of the crossing number of composite knots', arXiv:2407.10467 (v1 15 July 2024; v2 31 July 2026), https://arxiv.org/abs/2407.10467
4. A. Kawauchi, 'The crossing numbers of knots and links via 4D topology', arXiv:2608.13789 (13 August 2026), https://arxiv.org/abs/2608.13789
5. K. Murasugi, 'Jones polynomials and classical conjectures in knot theory', Topology 26 (1987), no. 2, 187–194. DOI: 10.1016/0040-9383(87)90058-9, https://doi.org/10.1016/0040-9383(87)90058-9
6. T. Ito, 'A quantitative Birman–Menasco finiteness theorem and its application to crossing number', Journal of Topology 15 (2022), no. 4, 1794–1806. DOI: 10.1112/topo.12259, https://doi.org/10.1112/topo.12259
7. Y. Diao, 'The additivity of crossing numbers', Journal of Knot Theory and Its Ramifications 13 (2004), no. 7, 857–866. DOI: 10.1142/s0218216504003524, https://doi.org/10.1142/s0218216504003524