Quasi-additivity of the Delta-unknotting number, a strand-independent braid-knot constant, and primality of knots with Delta-unknotting number one

The Delta-unknotting number $u_\Delta(K)$ counts the minimum number of Delta-moves (clasp-pass moves) needed to deform a knot $K$ into the unknot. Baader and Lewark (2026) proved that $u_\Delta$ is quasi-additive on knots arising as closures of positive or negative braids on a fixed number $m$ of strands, with constant $1/(48m)$ , as a consequence of a new lower bound $a_2(K)\ge g^2/(6m)$ on the Casson invariant of positive braid knots of genus $g$ . This record collects the three questions the same paper leaves open, which together ask for the coarse behavior of $u_\Delta$ under connected sum: (1) does there exist a constant $c>0$ , valid for all knots, with $u_\Delta(K_1\#K_2)\ge c\,(u_\Delta(K_1)+u_\Delta(K_2))$ ; (2) does $u_\Delta(K)=1$ imply that $K$ is prime, the Delta-analogue of Scharlemann's theorem for the ordinary unknotting number; and (3) can the constant $1/(48m)$ be replaced by one independent of $m$ , which would make $u_\Delta$ quasi-additive on the entire class of positive and negative braid knots. None of the three questions has been resolved or partially settled in the literature.

Background

A knot is a smooth embedding of a circle in the 3-sphere, considered up to continuous deformation (ambient isotopy). The connected sum $K_1\#K_2$ of two knots is formed by cutting each open and gluing the ends together in the standard way, and a knot is prime if it cannot be written as a connected sum of two nontrivial knots. The ordinary unknotting number $u(K)$ is the minimum number of crossing changes (passing one strand through another) needed to turn $K$ into the unknot. The Delta-move ( $\Delta$ -move), equivalent to the clasp-pass move, is a different local move on a knot diagram that changes three crossings in a triangular configuration simultaneously; Murakami and Nakanishi (1989) proved that Delta-moves alone suffice to unknot any knot, so the Delta-unknotting number $u_\Delta(K)$ — the minimum number of Delta-moves needed to reach the unknot — is well defined for every knot.

The strongest general lower bound for $u_\Delta$ is the Casson invariant $a_2(K)$ , the coefficient of $z^2$ in the Conway polynomial $\nabla_K(z)=1+a_2(K)z^2+a_4(K)z^4+\dots$ (a classical polynomial knot invariant computable from any diagram), which changes by at most one under each Delta-move, giving Okada's inequality $|a_2(K)|\le u_\Delta(K)$ . Since $a_2$ is additive under connected sum, this bound implies that $u_\Delta$ is exactly additive on all knots with up to seven crossings, and it is sharp on torus knots (knots lying on the surface of a standardly embedded torus, such as the trefoil, the simplest nontrivial knot), positive pretzel knots (knots formed by connecting several twisted bands end to end in a cycle), and closures of positive 3-braids (Nakamura, Nakanishi, Uchida, 1998), making potential counterexamples to additivity hard to find.

Coarse additivity questions became urgent in 2025, when Brittenham and Hermiller disproved additivity of the ordinary unknotting number by exhibiting $u(7_1\#\overline{7_1})\le 5<6$ , where $7_1$ is the seven-crossing knot of the standard tables and $\overline{7_1}$ its mirror image. An invariant is called quasi-additive if there is a universal constant $c>0$ with $f(K_1\#K_2)\ge c\,(f(K_1)+f(K_2))$ for all knots, a property Lackenby established for the crossing number in 2009; for both $u$ and $u_\Delta$ , quasi-additivity is open. Baader and Lewark (2026) proved the first partial result for $u_\Delta$ : writing $\widehat{B_m^+}$ and $\widehat{B_m^-}$ for the sets of links obtained as closures of positive resp. negative braids on $m$ strands (positive braids being finite products of the standard braid generators $\sigma_1,\dots,\sigma_{m-1}$ ), they showed

$u_\Delta(K_1\#K_2)\ \ge\ \frac{1}{48m}\bigl(u_\Delta(K_1)+u_\Delta(K_2)\bigr)$

for all knots $K_1,K_2\in\widehat{B_m^+}\cup\widehat{B_m^-}$ . The proof rests on their new estimate $g^2/(6m)\le a_2(K)\le g(g+1)/2$ for closures of positive $m$ -braids of minimal Seifert genus $g$ (the Seifert genus of a knot being the smallest genus of an oriented surface spanning it; the upper bound is due to Van Buskirk), whose lower bound is sharp up to a factor of six: the $n$ -fold connected sum of trefoils is the closure of the positive braid $\sigma_1^3\sigma_2^3\cdots\sigma_n^3$ , has genus $n$ and $a_2=n$ . For $m=3$ the Casson bound is attained, so $u_\Delta$ is genuinely additive on positive and negative 3-braid knots. The present record concerns the questions the authors pose at the close of the same paper. The primality question mirrors Scharlemann's 1985 theorem that $u(K)=1$ forces $K$ to be prime; for $u_\Delta$ , every knot currently known or conjectured to have $u_\Delta=1$ — the entries of Okada's table (as later corrected), the two-bridge families studied by Nakamura (two-bridge knots being the rational knots, so called because they admit a height function with exactly two maxima), and Higa's infinite family of trefoil-like knots — is prime, but no argument beyond the prime-ness of those particular families is known.

Problem Statement

The problem is to determine the coarse behavior of the Delta-unknotting number $u_\Delta$ under connected sum, as posed by Baader and Lewark (arXiv:2607.20212): questions (1) and (2) appear in the closing remarks of that paper, and question (3) is raised in its introduction as a possible strengthening of its Theorem 1. Three precise questions constitute the target.

(1) Quasi-additivity for all knots. Does there exist a constant $c>0$ such that

$u_\Delta(K_1\#K_2)\ \ge\ c\,\bigl(u_\Delta(K_1)+u_\Delta(K_2)\bigr)$

holds for all knots $K_1,K_2$ in the 3-sphere?

(2) Primality at level one. Does $u_\Delta(K)=1$ imply that $K$ is prime — equivalently, is it impossible to unknot a connected sum of two nontrivial knots with a single Delta-move?

(3) Strand-independent braid-knot constant. Does there exist a constant $c>0$ , independent of the number of strands $m$ , such that the inequality of (1) holds for all closures $K_1,K_2$ of positive or negative braids of arbitrary braid index (the braid index being the number of strands of the braid)? This would strengthen Theorem 1 of the source, which gives the strand-dependent constant $1/(48m)$ on $\widehat{B_m^+}\cup\widehat{B_m^-}$ for each fixed $m$ , to the entire class of positive and negative braid knots.

The three questions share a single scientific objective — the connected-sum behavior of $u_\Delta$ — with (3) the natural strengthening of the source's theorem toward (1), and (2) the qualitative level-one case of the same behavior.

References

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