Does the unknotting number of a connected sum of knots grow with the number of summands?

For $n\ge 2$ nontrivial knots $K_1,\dots,K_n$ in the 3-sphere, the unknotting number $u(K_1\#\cdots\#K_n)$ is known to be at least $2$ (Scharlemann: unknotting-number-one knots are prime), but no general lower bound that grows with $n$ is known: it is open whether $u(K_1\#\cdots\#K_n)\ge n$, and even whether the connected sum of three nontrivial knots must have unknotting number greater than $2$. The problem asks for lower bounds on the unknotting number of multi-summand connected sums — growth of $u$ with the number of summands in full generality, including summands with trivial or pairwise coprime Alexander polynomials — and whether the classical algebraic lower bounds, Nakanishi's invariant $m(K)$ and the Ma–Qiu index $a(K)$ (with $u\ge a\ge m$), are additive under connected sum, which is itself open. The June 2025 counterexample of Brittenham and Hermiller ($u(7_1\#\overline{7_1})\le 5<6$) refuted the additivity conjecture (Kirby problem 1.69(B)), making the growth question the central open problem about unknotting numbers of composite knots.

Background

A knot $K$ is an embedded circle in the 3-sphere $S^3$, considered up to continuous deformation. The unknotting number $u(K)$ is the minimum number of crossing changes (switching which strand passes over at a crossing of a diagram, with arbitrary deformation allowed between changes) needed to turn $K$ into the unknot. It is one of the oldest measures of knotting complexity, yet remains notoriously hard to compute: as of 2025, nine knots with at most ten crossings — all with crossing number exactly ten — still have unknown unknotting number.

The connected sum $K_1\#K_2$ of two oriented knots is formed by cutting a short arc out of each and gluing the four endpoints pairwise; a knot is prime if it is not a connected sum of two nontrivial knots, and every knot factors uniquely into prime knots (Schubert). Most classical invariants behave simply under connected sum — knot polynomials are multiplicative, the Seifert genus (the minimal genus of an oriented surface whose boundary is the knot) and the signature $\sigma$ (an integer-valued invariant satisfying $|\sigma(K)|\le 2u(K)$) are additive — and it was long conjectured, appearing in problem lists since the 1970s and as Problem 1.69(B) in Kirby's list, that unknotting number is additive too: $u(K_1\#K_2)=u(K_1)+u(K_2)$. The upper bound $u(K_1\#K_2)\le u(K_1)+u(K_2)$ is trivial; all the difficulty lies below.

The strongest general lower bound is Scharlemann's 1985 theorem that a knot with $u(K)=1$ is prime; hence the connected sum of any two nontrivial knots has $u\ge 2$, and additivity does hold when both summands have unknotting number $1$. The signature gives additivity when the summands' signatures have the same sign and realize $|\sigma|=2u$. Beyond this, the classical strategy is algebraic. The Alexander polynomial $\Delta_K(t)$ is computed from the homology $H_1$ of the infinite cyclic cover of the knot complement, a module over the Laurent polynomial ring $\mathbb{Z}[t^{\pm1}]$ (the Alexander module). Nakanishi (1981) defined $m(K)$ as the minimal number of generators of this module and proved $u(K)\ge m(K)$; Ma and Qiu (2006) defined $a(K)$ as the minimal number of elements of the commutator subgroup of the knot group $\pi_1(S^3\setminus K)$ whose normal closure (the smallest normal subgroup containing them) is the whole commutator subgroup, and proved $u(K)\ge a(K)\ge m(K)$. Unlike $m$, which vanishes for the nontrivial knots with $\Delta_K=1$, the invariant $a$ is at least $1$ for every nontrivial knot. Under connected sum, however, only $a(K_1\#K_2)\ge\max\{a(K_1),a(K_2)\}$ is known; additivity of $a$ or of $m$ is open. Yang (2008) proved that the connected sum of $n$ copies of a knot with $\Delta_K\ne 1$ has $u\ge n$, using the count of invariant factors of the Alexander module tensored with $\mathbb{Q}$, which adds for identical summands; for distinct summands the same argument needs their Alexander polynomials to share a prime factor, since cyclic modules with coprime orders combine under direct sum. Two further lower bounds complete the algebraic toolkit: the algebraic unknotting number $u_a(K)$ — the minimum number of crossing changes needed to turn $K$ into a knot with trivial Alexander polynomial, so $u_a(K)\le u(K)$ — which Borodzik and Friedl proved to be exactly computable from the Blanchfield pairing, a sesquilinear form on the Alexander module; and the knot Floer torsion order $\mathrm{Ord}(K)$, an algebraic invariant of knot Floer homology (a categorification of the Alexander polynomial defined from Heegaard diagrams), which satisfies $\mathrm{Ord}(K)\le u(K)$ by work of Zhuang. Neither is known to be additive under connected sum.

This landscape was upended in June 2025, when Brittenham and Hermiller refuted the additivity conjecture: for the $(2,7)$-torus knot $7_1$ (a knot wrapping twice around one direction and seven around the other of a standard torus; $u(7_1)=3$) they gave an explicit 5-crossing-change unknotting sequence for $7_1\#\overline{7_1}$, the sum of $7_1$ with its mirror image, so $u(7_1\#\overline{7_1})\le 5<6$ (Wang and Zhang gave an independent verification). Non-additivity now comes in infinite families, with arbitrarily large gaps, and even in a 'symbiosis' form: $u(4_1\#9_{10})\le 3=u(9_{10})$, so summing with the figure-eight knot $4_1$ does not raise the unknotting number at all. Additivity of unknotting number is therefore dead; what survives, as Brittenham and Hermiller themselves emphasize, is exactly the question of how small the unknotting number of a multi-summand sum can be: their Question 4.2 asks whether $u(K\#K')\ge\max\{u(K),u(K')\}$ always holds, and the exact value of $u(7_1\#\overline{7_1})$ (known only to lie between $2$ and $5$) is open. The present problem is the multi-summand growth question at the heart of this post-2025 landscape.

Problem Statement

For nontrivial knots $K_1,\dots,K_n$ in $S^3$, determine how the unknotting number of their connected sum grows with the number of summands. The objective is a general lower bound for $u(K_1\#\cdots\#K_n)$ that grows with $n$, or an explicit family whose unknotting number stays bounded as the number of summands grows. Sub-question 1 is the general lower bound, Sub-question 2 a weaker consequence of it, and Sub-question 3 an independent algebraic direction.

1. Is $u(K_1\#\cdots\#K_n)\ge n$ for all $n\ge 1$ and all choices of nontrivial knots — equivalently, does every nontrivial summand force at least one crossing change? In particular, must the connected sum of three nontrivial knots have unknotting number greater than $2$? This is open even when all summands have trivial Alexander polynomial ($\Delta=1$) — then the Alexander-module-based bounds $m$ and $u_a$ vanish, and no nonzero signature, hence no signature-based lower bound, is known for such knots — and when the summands' Alexander polynomials are pairwise coprime and their signatures have mixed signs, so the additive signature bound cancels: these are the regimes in which the module-theoretic bounds ($m$, $a$, $u_a$) vanish or collapse to the maximum over summands, and no published lower bound for the composite exceeds $u\ge 2$.

2. A weaker consequence of (1): does $u(K_1\#\cdots\#K_n)\to\infty$ as $n\to\infty$ — for instance for the iterated sum $K^{\# n}$ of a fixed knot $K$ with $\Delta_K=1$, for which Yang's theorem gives nothing? A negative answer would be an explicit family of multi-summand knots whose unknotting number stays bounded as the number of summands grows; a positive answer for a single family such as $K^{\# n}$ would not by itself settle the general question. Closely related: is $u(K\#K')\ge\max\{u(K),u(K')\}$ for all knots (Brittenham–Hermiller Question 4.2), and what is the exact value of the emblematic open case $u(7_1\#\overline{7_1})$, currently known only to lie between $2$ and $5$?

3. Can the known lower bounds be made to grow under connected sum — specifically, is $a(K_1\#\cdots\#K_n)\ge\sum_i a(K_i)$ — additivity of the Ma–Qiu index, a natural question because the knot group of a composite knot is the free product (the group-theoretic analogue of disjoint union) of the summands' knot groups — or the analogous statement for Nakanishi's $m$ or for the algebraic unknotting number $u_a$; and, where they cannot, how large are the gaps $u-a$ and $a-m$ for multi-summand connected sums?

References

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