Exact unknotting numbers of the connected sums $T(2,k)\,\#\,T(2,-k)$ of two-strand torus knots with their mirrors, and exact Reidemeister move counts for unlinking

Hayashi, Hayashi and Nowik (Topology and its Applications 159, 2012) proved that certain diagrams $D_n$ of the trivial two-component link require a number of Reidemeister moves growing quadratically in the crossing number to be unknotted, and — conditionally on their Conjecture 1.7, that the unknotting number satisfies $u(T(2,k)\,\#\,T(2,-k))=k-1$ for every odd $k$, where $T(2,k)$ is the two-strand torus knot and $T(2,-k)$ its mirror — obtained corresponding closed-form bounds for splitting. The even-$k$ cases were settled in the paper itself ($u=k$), and $k=3$ (the square knot, $u=2$) was already classical. Brittenham and Hermiller (2025) then proved that unknotting number is not additive under connected sum, and showed $u(T(2,7)\,\#\,T(2,-7))\le 5<6$ and, more generally, $u(T(2,k)\,\#\,T(2,-k))\le k-2$ for every odd $k\ge 7$: Conjecture 1.7 is false, and the conditional sharp move-count formulas are unattainable for $n\ge 7$. The surviving open core is the exact determination of $u(T(2,k)\,\#\,T(2,-k))$ for odd $k\ge 5$ (for $k=5$ the value lies between 2 and 4, with 4 the additive prediction; for $k\ge 7$ it lies between 2 and $k-2$), together with the exact minimal numbers of matched RII and RIII moves needed to split the diagrams $D_n$.

Background

A link diagram is a planar projection of a link with over/under information at each double point. Reidemeister's theorem states that two diagrams of the same link are always connected by a finite sequence of local moves of three types: RI (creating or deleting a monogon face), RII (creating or deleting a bigon face) and RIII (replacing one triangle by another); an RII move is called matched or unmatched according to whether the orientations of the two edges of the bigon face agree or oppose. A natural quantitative question, pioneered by Hass and Nowik, is how many such moves specific diagrams actually need. The unknotting number $u(L)$ of a link $L$ is the minimal number of crossing changes (interchanging the over- and under-strand at crossings), with isotopy allowed between them, needed to transform $L$ into the trivial link with the same number of components. The connected sum $K_1\,\#\,K_2$ of knots is formed by deleting a short arc from each and reconnecting the endpoints; a knot that is such a sum of two nontrivial knots is called composite. Always $u(K_1\,\#\,K_2)\le u(K_1)+u(K_2)$; the reverse inequality for all pairs was the long-standing additivity conjecture (Problem 1.69(B) in Kirby's problem list). The $(2,k)$-torus knot or link $T(2,k)$, the closure of the 2-braid $\sigma_1^k$ (that is, of $k$ half-twists), is a knot for odd $k$ and a 2-component link for even $k$; a theorem of Kronheimer and Mrowka (proving a conjecture of Milnor) gives $u(T(p,q))=(p-1)(q-1)/2$, so $u(T(2,k))=(k-1)/2$ for odd $k$. The linking number of two oriented link components is the integer measuring how they interlink; a crossing change between two components alters it by one, so the sum of absolute linking numbers bounds the unlinking number from below. Hayashi, Hayashi and Nowik built a link-diagram invariant $iu(D)$ of Hass–Nowik type: it sums, over all crossings, the sign of the crossing times the absolute value of the difference between the unknotting number of the link obtained by oriented smoothing at that crossing and the unknotting number of the link itself. It does not change under RI or unmatched RII moves and changes by at most 1 under a matched RII move and at most 2 under an RIII move, so $|iu(D_1)-iu(D_2)|/2$ lower-bounds the number of matched RII and RIII moves between two diagrams. For the diagrams $D_n$ — closures of the explicit $(n+1)$-braid $\sigma_1^{-1}(\sigma_2^{-1}\sigma_1^{-1})\cdots(\sigma_n^{-1}\sigma_{n-1}^{-1})\sigma_n^n$ on the trivial two-component link, with $3n-1$ crossings, where $T(2,-k)$ denotes the mirror image of $T(2,k)$ — they proved that unknotting requires a number of moves bounded below by an explicit sum of the numbers $u(T(2,k)\,\#\,T(2,-k))$, and, via a splitting variant $iu'$, that splitting $D_n$ requires a number of matched RII and RIII moves bounded below by sums over odd $k$ of the same quantities. For even $k$, where $T(2,k)\,\#\,T(2,-k)$ is a 3-component link, they proved $u=k$ by linking numbers. For odd $k$ the summand is a composite knot (for $k=3$, the square knot, i.e. a trefoil summed with its mirror, with $u=2$ by Scharlemann's theorem that unknotting-number-one knots are prime, together with the evident two-change sequence), and they could only bound $u\ge 2$; they posed Conjecture 1.7: $u(T(2,k)\,\#\,T(2,-k))=k-1$ for every odd $k$, the value predicted by additivity. Under this conjecture their splitting lower bounds become the exact closed forms $n(n-2)/4$ ($n$ even) and $(n-1)^2/4$ ($n$ odd). The additivity landscape has since changed dramatically. Brittenham and Hermiller (2025) proved that unknotting number is not additive under connected sum: $u(7_1\,\#\,\overline{7_1})\le 5<6$, where $7_1=T(2,7)$ and $\overline{K}$ denotes the mirror image of $K$, and $u(T(2,2k+1)\,\#\,\overline{T(2,2l+1)})<u(T(2,2k+1))+u(T(2,2l+1))$ for all $k,l\ge 3$; Wang and Zhang independently confirmed the $7_1$ case with an explicit 56-crossing diagram unknotted by 5 crossing changes. In 2026 they showed $u(4_1\,\#\,9_{10})\le 3<1+3$ and $u(5_1\,\#\,\overline{8_2})\le 3<2+2$, and that every non-trivial torus knot except possibly the trefoil has a symbiont, i.e. a partner $K'$ with $u(T\,\#\,K')<u(T)+u(K')$. On the other hand, no lower-bound technique currently reaches the additive values on the mirror-pair family: the signature (a classical integer-valued concordance invariant, satisfying $|\sigma|\le 2u$), the Levine–Tristram signature function (the family of signatures obtained from the Seifert form evaluated at points on the unit circle), and the homological concordance invariants $\tau$ and $s$ (from knot Floer homology and Khovanov homology respectively, each bounded above by the unknotting number through the slice genus, i.e. the minimal genus of a surface the knot bounds in the four-ball) all vanish on $K\,\#\,\overline{K}$, and the algebraic unknotting number of Borodzik and Friedl — the minimum number of crossing changes needed to trivialize the Seifert matrix (the linking matrix of a Seifert surface for the knot) up to S-equivalence, which subsumes every lower bound derived from it — is itself at most $u$, hence at most $k-2$ for odd $k\ge 7$.

Problem Statement

Primary question: for each odd integer $k\ge 5$, determine the exact unknotting number $u(T(2,k)\,\#\,T(2,-k))$ of the connected sum of the $(2,k)$-torus knot with its mirror. This is the surviving core of Hayashi–Hayashi–Nowik's Conjecture 1.7, which asserted $u(T(2,k)\,\#\,T(2,-k))=k-1$ for every odd $k$ and is now known to be false for every odd $k\ge 7$ (where in fact $u\le k-2$). Concretely: (1) decide whether $u(T(2,5)\,\#\,T(2,-5))=4$, the value predicted by additivity — currently only $2\le u\le 4$ is known; (2) for each odd $k\ge 7$, determine the exact value, currently only known to satisfy $2\le u\le k-2$, including whether the values grow linearly in $k$, match $k-2$, or behave otherwise. Linked derived question from the same source: (3) determine the exact minimal number of matched RII and RIII moves needed to deform the diagrams $D_n$ of the trivial two-component link (specified by their braid words in the source, diagrams drawn on $S^2$) into a split diagram: the Hayashi–Hayashi–Nowik lower bounds are explicit sums of the numbers $u(T(2,k)\,\#\,T(2,-k))$ and evaluate to the closed forms $n(n-2)/4$ ($n$ even) and $(n-1)^2/4$ ($n$ odd) precisely when the additive values hold — which is now known to fail for every $n\ge 7$ — so both the true minimal counts and the tightness of the existing bounds remain to be established.

References

1. Chuichiro Hayashi, Miwa Hayashi, Tahl Nowik, 'Unknotting number and number of Reidemeister moves needed for unlinking', Topology and its Applications 159 (2012) 1467-1474, DOI 10.1016/j.topol.2012.01.008, https://doi.org/10.1016/j.topol.2012.01.008; preprint arXiv:1012.4131 (the arXiv version lists only the first two authors), https://arxiv.org/abs/1012.4131
2. Mark Brittenham, Susan Hermiller, 'Unknotting number is not additive under connected sum', preprint arXiv:2506.24088 (v2, September 2025), https://arxiv.org/abs/2506.24088
3. Mark Brittenham, Susan Hermiller, 'Unknotting number and connected sums: The knots $4_1$ and $5_1$', preprint arXiv:2601.18757 (January 2026), https://arxiv.org/abs/2601.18757
4. Martin Scharlemann, 'Unknotting number one knots are prime', Inventiones Mathematicae 82 (1985) 37-55, DOI 10.1007/BF01394778, https://doi.org/10.1007/BF01394778
5. Peter Kronheimer, Tomasz Mrowka, 'Gauge theory for embedded surfaces, I', Topology 32 (1993) 773-826, DOI 10.1016/0040-9383(93)90051-V, https://doi.org/10.1016/0040-9383(93)90051-V
6. Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, Daniel Zheng, 'The unknotting number, hard unknot diagrams, and reinforcement learning', Experimental Mathematics, published online 2025, pp. 1-19, DOI 10.1080/10586458.2025.2542174, https://doi.org/10.1080/10586458.2025.2542174; preprint arXiv:2409.09032, https://arxiv.org/abs/2409.09032
7. Maciej Borodzik, Stefan Friedl, 'On the algebraic unknotting number', Transactions of the London Mathematical Society 1 (2014) 57-84, DOI 10.1112/tlms/tlu004, https://doi.org/10.1112/tlms/tlu004; preprint arXiv:1308.6105, https://arxiv.org/abs/1308.6105
8. Chao Wang, Yimu Zhang, 'A remark on the counterexample to the unknotting number conjecture', preprint arXiv:2507.14265 (2025), https://arxiv.org/abs/2507.14265