How badly unknotting number fails to be additive under connected sum: the max bound, universally additive knots, and the exact value of $u(7_1\#\overline{7_1})$

Brittenham and Hermiller (2025) disproved the century-old conjecture that the unknotting number $u$ is additive under connected sum by proving $u(7_1\#\overline{7_1})\le 5<6=u(7_1)+u(\overline{7_1})$, where $7_1$ is the $(2,7)$-torus knot; by 2026 they had extended non-additivity to $4_1\#9_{10}$ and $5_1\#8_2$. This task targets the surviving open core of the unknotting-number additivity problem: (a) decide whether $u(K\#K')\ge\max\{u(K),u(K')\}$ holds for all knots — the exact analogue already fails for the band-unknotting number; (b) decide whether any non-trivial knot is universally additive, i.e. $u(K\#K')=u(K)+u(K')$ for every partner $K'$, in direct competition with the authors' 2026 'symbiont' anti-conjecture asserting that every non-trivial knot has a partner violating additivity; and (c) determine the exact value of $u(7_1\#\overline{7_1})$, currently known only to lie between $2$ and $5$.

Background

A knot $K$ is a smoothly embedded circle in the 3-sphere $S^3$, and its unknotting number (or Gordian number) $u(K)$ is the minimum number of crossing changes — switches of an over-strand to an under-strand at a crossing of a planar diagram, with isotopies of the diagram allowed in between — needed to turn $K$ into the unknot. The connected sum $K\#K'$ is formed by deleting a short arc from each of two disjoint knot diagrams and reconnecting the endpoints; a knot is prime if it is not a connected sum of two non-trivial knots, and by Schubert's theorem every knot factors uniquely into prime knots, so many knot-theoretic questions reduce to prime knots. The mirror image $\overline{K}$ is the reflection of $K$, and $u(\overline{K})=u(K)$; a torus knot $T(p,q)$ lies on the surface of a standard donut, wrapping $p$ times one way and $q$ the other, and Kronheimer and Mrowka proved, using gauge theory (invariants of the knot complement built from the differential equations of mathematical physics), that $u(T(p,q))=(p-1)(q-1)/2$.

Many classical invariants are additive under connected sum — among them the genus (the minimal genus of an oriented spanning surface for the knot), the signature $\sigma(K)$ (the signature of the symmetrized intersection pairing on such a spanning surface) — and it was long conjectured — implicitly in Wendt's 1937 work and explicitly in Gordon's 1977 and Kirby's problem lists (problem 1.69(B)) — that $u(K\#K')=u(K)+u(K')$. The inequality $u(K\#K')\le u(K)+u(K')$ is trivial (unknot each summand), so the conjecture was the reverse inequality. Evidence in its favor: Scharlemann (1985) proved that knots with $u=1$ are prime, so the connected sum of two non-trivial knots has $u\ge 2$; and Murasugi's signature bound $|\sigma(K)|\le 2u(K)$ together with additivity of the signature $\sigma$ gives full additivity whenever $|\sigma|=2u$ for both summands with matching sign. The only other general lower bounds for composite knots are of this additive type and therefore vanish on mirror pairs, whose signature is $0$.

In 2025 Brittenham and Hermiller disproved the conjecture: for $K=7_1=T(2,7)$ with $u(K)=3$, they exhibited a diagram of $K\#\overline{K}$ and a sequence of $5$ crossing changes through the intermediate knots $K14a18636$ and $K15n81556$ that reaches the unknot, so $u(7_1\#\overline{7_1})\le 5<6$ (Wang and Zhang later corrected a chirality slip in one intermediate diagram of v1 and gave a direct verification; the bound stands). Their mechanism is Gordian adjacency — knot $K_1$ is Gordian adjacent to $K_2$ if $K_1$ appears in a minimal unknotting sequence for $K_2$ — which bootstraps one example into infinite families (e.g. all $T(2,2k+1)\#\overline{T(2,2\ell+1)}$ for $k,\ell\ge 3$) with arbitrarily large additivity gap. A byproduct is that the knot group $\pi_1(S^3\setminus K)$ alone does not determine $u(K)$.

A 2026 follow-up by the same authors shows the phenomenon reaches the simplest knots: $u(4_1\#9_{10})\le 3$ where $u(4_1)=1$ and $u(9_{10})=3$ (the latter due to Owens via Heegaard Floer homology, a modern knot invariant built by counting holomorphic disks in an auxiliary space) — so summing with $4_1$ does not raise the unknotting number at all — and $u(5_1\#8_2)\le 3<4$ with mirroring chosen so the signature cancels; they call a pair with $u(K\#K')<u(K)+u(K')$ symbionts, and prove every non-trivial torus knot except possibly the trefoil has one. They also show $u(3_1\#10_6)\le 3$, which gives the trefoil a symbiont exactly if the still-open value $u(10_6)\in\{2,3\}$ equals $3$. Reinforcement-learning searches (Applebaum et al.; Dranowski–Kabkov–Tubenhauer) both preceded and reproduced these upper bounds, but produced no lower bounds. Ghanbarian and Jabuka proved the analogous failure is stronger for the band-unknotting number (the minimum number of band surgeries, a related operation): there, infinitely many pairs satisfy $u_{nb}(K_1\#K_2)<u_{nb}(K_i)$, i.e. the max bound fails outright. Whether the same can happen for the classical unknotting number — Brittenham–Hermiller's Question 4.2 — is open, as are Question 4.3 (existence of a universally additive non-trivial knot, negated by the authors' 2026 'symbiont' Conjecture 5.1) and Question 4.4 (the exact value of $u(7_1\#\overline{7_1})$). Brittenham–Hermiller also pose, as a separate algebraic-computability thread, how the 2-generalized knot group $G_2(K)$ — the fundamental group of the complement of $K$ together with two chosen meridians, which by Nelson–Neumann determines $K$ up to mirroring — behaves under connected sum and whether $u$ is computable from it.

Problem Statement

Determine how far unknotting number additivity can fail at the level of a single connected sum, by resolving the following three linked questions of Brittenham–Hermiller (arXiv:2506.24088, Questions 4.2–4.4; sharpened in arXiv:2601.18757). Here $u(K)$ is the unknotting number of a knot $K\subset S^3$, $K\#K'$ the connected sum, and $\overline{K}$ the mirror image, with $u(\overline{K})=u(K)$.

(A) Decide whether $u(K\#K')\ge\max\{u(K),u(K')\}$ for all knots $K$ and $K'$: either prove this max bound in general, or exhibit an explicit pair with $u(K\#K')<\max\{u(K),u(K')\}$. Currently the only general lower bound for a sum of non-trivial knots is $u\ge 2$ (Scharlemann), and the sharp test case is $u(4_1\#9_{10})\in\{2,3\}$ with $u(4_1)=1$, $u(9_{10})=3$: the value $2$ refutes the bound.

(B) Decide whether there exists a non-trivial knot $K$ that is universally additive, i.e. $u(K\#K')=u(K)+u(K')$ for every knot $K'$: either exhibit and certify such a knot, or prove the competing 2026 'symbiont' conjecture that every non-trivial knot $K$ has some partner $K'$ with $u(K\#K')<u(K)+u(K')$ (equivalently, that no non-trivial universally additive knot exists).

(C) Determine the exact value of $u(7_1\#\overline{7_1})$, where $7_1$ is the $(2,7)$-torus knot: currently $2\le u(7_1\#\overline{7_1})\le 5$, the upper bound from Brittenham–Hermiller's explicit 5-crossing-change sequence and the lower bound from Scharlemann's theorem.

These three parts share one scientific objective — quantifying the failure of additivity of the most classical knot invariant under the most basic knot-composition operation — and each is a decision or determination with its own recognizable completion boundary; the objective is complete when all three are decided.

References

1. Mark Brittenham and Susan Hermiller, Unknotting number is not additive under connected sum, arXiv:2506.24088 (v1 2025-06-30, v2 2025-09-15), DOI 10.48550/arxiv.2506.24088, https://arxiv.org/abs/2506.24088
2. Mark Brittenham and Susan Hermiller, Unknotting number and connected sums: The knots $4_1$ and $5_1$, arXiv:2601.18757 (2026), https://arxiv.org/abs/2601.18757
3. Chao Wang and Yimu Zhang, A remark on the counterexample to the unknotting number conjecture, arXiv:2507.14265 (2025), https://arxiv.org/abs/2507.14265
4. Nakisa Ghanbarian and Stanislav Jabuka, Band-unknotting numbers and connected sums of knots, arXiv:2512.06299 (2025), https://arxiv.org/abs/2512.06299
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