The band-unknotting number of the double twist knots $C(22+8k,62+8k)$

The band-unknotting number $u_{nb}(K)$ of a knot $K$ is the minimum number of non-orientable band moves needed to turn a diagram of $K$ into a diagram of the unknot. Ghanbarian and Jabuka (2025) proved that $u_{nb}$ is not additive under connected sum and can even drop: for $K_1=C(22+8k,62+8k)$ and $K_2=-\bar K_1$ one has $u_{nb}(K_1\#K_2)=2$ while $u_{nb}(K_i)\ge 3$. The exact band-unknotting numbers of these double twist knots are, however, unknown: the source paper states 'We don't know the band-unknotting number of $C(22+8k,62+8k)$', and only the bounds $3=\gamma_{4,s}\le u_{nb}(C(22+8k,62+8k))\le 7+2k$ are available. The open problem is to determine $u_{nb}(C(22+8k,62+8k))$ exactly for every integer $k\ge 0$, within the broader bottleneck of computing $u_{nb}$ and its lower-bound invariants for double twist knots; even the first member $C(22,62)$ is undetermined between 3 and 7.

Background

A knot is a smooth embedding of a circle into the 3-sphere $S^3$, studied up to continuous deformation (isotopy); the unknot is the knot that bounds an embedded disk. An unknotting operation is a local move on knot diagrams that can convert any knot into the unknot after finitely many applications, and the associated unknotting number is the minimum number of such moves, taken over all diagrams of the knot. The classical (Gordian) unknotting number $u(K)$ uses the crossing change, which switches one over-strand to an under-strand. This record concerns the non-orientable band move: a flat band $[0,1]^2$ is attached along two disjoint arcs of a knot diagram so that exactly one of the band's two attaching edges agrees with the orientation of its attaching arc, inherited from an orientation of the knot, while the other attaching edge disagrees (equivalently, the band carries a half-twist relative to the framing induced by the knot), and the two attaching arcs are then replaced by the other two sides of the band. Because a crossing change decomposes into two non-orientable band moves, these moves form an unknotting operation, and the band-unknotting number $u_{nb}(K)$ is the minimum number of them that unknots $K$. It coincides with the $H(2)$-unknotting number of Hoste, Nakanishi and Taniyama, where $H(2)$ is the second of their family of tangle-replacement moves.

The connected sum $K_1\#K_2$ splices two knots into one, and every such unknotting number is subadditive: $u(K_1\#K_2)\le u(K_1)+u(K_2)$, and likewise for $u_{nb}$. Whether the Gordian unknotting number is actually additive was a long-standing open question until Brittenham and Hermiller (2025) produced the first counterexample, $u(7_1\#\bar 7_1)\le 5<6$, where $\bar K$ denotes the mirror image. They also asked whether at least the weaker monotonicity $u(K_1\#K_2)\ge\max\{u(K_1),u(K_2)\}$ holds for all knots; that question concerns the classical invariant, remains open, and is separate from the present problem.

Ghanbarian and Jabuka (2025) established the band analogues: $u_{nb}$ is not additive, and it even fails the max-inequality, with infinitely many pairs $K_1,K_2$ satisfying $u_{nb}(K_1\#K_2)<u_{nb}(K_i)$ for $i=1,2$. Their construction is built from the double twist knots $C(m,n)$: two-bridge knots (knots of bridge index two, meaning they admit a diagram with only two local maxima) formed from two vertical twisting regions holding $m$ and $n$ half-twists between a pair of parallel strands; $C(m,n)$ is a knot exactly when at least one of $m,n$ is even. For these knots several bounds combine. The smooth (respectively topological) non-orientable 4-genus $\gamma_{4,s}(K)$ (respectively $\gamma_{4,t}(K)$) is the minimum first Betti number of a non-orientable surface, smoothly (respectively topologically flatly) and properly embedded in the 4-ball, whose boundary is $K$; Ghanbarian-Jabuka prove the chain $\mu_{an}(K)\le\gamma_{4,t}(K)\le\gamma_{4,s}(K)\le u_{nb}(K)$, where $\mu_{an}(K)$ is the minimal number of generators of the anisotropic part of the linking form - the canonical $\mathbb{Q}/\mathbb{Z}$-valued symmetric pairing, given by linking numbers, on the torsion group $H_1(\Sigma_2(K))$, with $\Sigma_2(K)$ the two-fold cyclic cover of $S^3$ branched along $K$ - using the orthogonal decomposition of such forms into an anisotropic summand (on which every nonzero element has nonzero self-pairing) and a metabolic, hyperbolic summand. Separately, the fusion number $F(R)$ of a ribbon knot $R$ is the fewest oriented bands that must be attached to $R$ to obtain an unlink; a bound of Juhász, Miller and Zemke gives $F(K\#(-\bar K))\le \mathrm{br}(K)-1$, with $\mathrm{br}$ the bridge index, and Lickorish's 1986 obstruction implies $u_{nb}(K\#(\pm\bar K))\ge 2$ whenever $\det K>1$.

Hoste, Shanahan and Van Cott determined the non-orientable 4-genus for many subfamilies of $C(m,n)$ using the Goeritz form (the integer intersection pairing on a spanning surface of the knot) and Donaldson's diagonalization theorem for definite intersection forms of smooth 4-manifolds, proving in their Theorem 6.2 that $\gamma_4(C(22+8k,62+8k))=3$ for all $k\ge 0$. Taking $K_1=C(22+8k,62+8k)$ and $K_2=-\bar K_1$ (the reverse of the mirror), Ghanbarian-Jabuka then obtain $u_{nb}(K_1\#K_2)=2<3\le u_{nb}(K_i)$, proving their non-monotonicity theorem. But the value of $u_{nb}$ on the family itself is left open: the paper states that the authors do not know the band-unknotting number of $C(22+8k,62+8k)$ and records only an elementary upper bound, so the state of knowledge is

$$3=\gamma_{4,s}\le u_{nb}(C(22+8k,62+8k))\le 7+2k,\qquad k\ge 0.$$

Existing lower-bound machinery is weak on this family: since $C(m,n)$ is two-bridge, $\Sigma_2(C(m,n))$ is a lens space - a 3-manifold built from two solid tori with cyclic first homology - so the Hoste-Nakanishi-Taniyama bound $\mu(K)\le u_{nb}(K)$, with $\mu(K)$ the minimal number of generators of $H_1(\Sigma_2(K))$, is nearly vacuous; Lickorish's obstruction only detects $u_{nb}=1$; and tabulations of band-unknotting numbers (Kanenobu-Miyazawa's $H(2)$-unknotting numbers, and Abe-Hanaki-Higa's computations) cover only small-crossing knots. Himeno (2025) determined, for torus knots, the analogous unoriented band-unknotting number - the variant allowing oriented band surgeries as well as non-orientable ones - proving it equals the pinch number there, i.e. the minimum number of Batson's pinch moves, a specific non-oriented band surgery on the standard torus-knot diagram, needed to reach the unknot; no analogous result exists for double twist knots.

Problem Statement

What is the band-unknotting number of the double twist knot $C(22+8k,62+8k)$ for each integer $k\ge 0$? Equivalently: close the gap between the lower bound $u_{nb}\ge\gamma_{4,s}=3$ of Hoste-Shanahan-Van Cott and the elementary upper bound $u_{nb}\le 7+2k$ of Ghanbarian-Jabuka, which already leaves the first member $C(22,62)$ undetermined between 3 and 7. Determining the value for every $k\ge 0$ means exhibiting, for each $k$, a finite sequence of non-orientable band moves on explicit diagrams of $C(22+8k,62+8k)$ whose end result is the unknot, and proving that no smaller number of non-orientable band moves unknots $C(22+8k,62+8k)$. The completion boundary here is the exact determination of $u_{nb}$ for the family $C(22+8k,62+8k)$, all $k\ge 0$.

References

1. Nakisa Ghanbarian and Stanislav Jabuka, Band-unknotting numbers and connected sums of knots, arXiv:2512.06299 (2025). DOI: 10.48550/arXiv.2512.06299, https://arxiv.org/abs/2512.06299 (source paper of this open problem)
2. Mark Brittenham and Susan Hermiller, Unknotting number is not additive under connected sum, arXiv:2506.24088 (2025). DOI: 10.48550/arXiv.2506.24088, https://arxiv.org/abs/2506.24088 (v1 June 2025, v2 September 2025)
3. Mark Brittenham and Susan Hermiller, Unknotting number and connected sums: The knots 4_1 and 5_1, arXiv:2601.18757 (2026). DOI: 10.48550/arXiv.2601.18757, https://arxiv.org/abs/2601.18757
4. Jim Hoste, Patrick D. Shanahan, and Cornelia A. Van Cott, On the nonorientable 4-genus of double twist knots, Journal of Knot Theory and Its Ramifications 32 (2023), no. 5, 2350036. DOI: 10.1142/s0218216523500360, https://arxiv.org/abs/2208.07850
5. W. B. R. Lickorish, Unknotting by adding a twisted band, Bulletin of the London Mathematical Society 18 (1986), no. 6, 613-615. DOI: 10.1112/blms/18.6.613, https://doi.org/10.1112/blms/18.6.613
6. Jim Hoste, Yasutaka Nakanishi, and Kouki Taniyama, Unknotting operations involving trivial tangles, Osaka Journal of Mathematics 27 (1990), no. 3, 555-566. Zbl 0713.57006, https://zbmath.org/?q=an:0713.57006
7. Taizo Kanenobu and Yasuyuki Miyazawa, H(2)-unknotting number of a knot, Communications in Mathematical Research 25 (2009), no. 5, 433-460. Zbl 1212.57001, https://zbmath.org/?q=an:1212.57001
8. Tetsuya Abe, Ryo Hanaki, and Ryuji Higa, The unknotting number and band-unknotting number of a knot, Osaka Journal of Mathematics 49 (2012), no. 2, 523-550. Zbl 1255.57002, https://zbmath.org/?q=an:1255.57002, https://projecteuclid.org/euclid.ojm/1340197938
9. Keisuke Himeno, The unoriented band unknotting numbers of torus knots, arXiv:2502.14304 (2025). DOI: 10.48550/arXiv.2502.14304, https://arxiv.org/abs/2502.14304
10. András Juhász, Maggie Miller, and Ian Zemke, Knot cobordisms, bridge index, and torsion in Floer homology, Journal of Topology 13 (2020), no. 4, 1701-1724. DOI: 10.1112/topo.12170, https://doi.org/10.1112/topo.12170
11. Chao Wang and Yimu Zhang, A remark on the counterexample to the unknotting number conjecture, arXiv:2507.14265 (2025). DOI: 10.48550/arXiv.2507.14265, https://arxiv.org/abs/2507.14265
12. Sebastian Baader and Lukas Lewark, Étude on the Delta-unknotting number, arXiv:2607.20212 (2026). DOI: 10.48550/arXiv.2607.20212, https://arxiv.org/abs/2607.20212
13. Hayato Imori, JungHwan Park, and Masaki Taniguchi, Unknotting number, ribbon concordance, and singular instantons, arXiv:2607.12768 (2026). DOI: 10.48550/arXiv.2607.12768, https://arxiv.org/abs/2607.12768
14. Duncan McCoy and JungHwan Park, Special alternating links of minimal unlinking number, arXiv:2603.10732 (2026). DOI: 10.48550/arXiv.2603.10732, https://arxiv.org/abs/2603.10732
15. Anne Dranowski, Yura Kabkov, and Daniel Tubbenhauer, RL unknotter, hard unknots and unknotting number, arXiv:2603.07955 (2026). DOI: 10.48550/arXiv.2603.07955, https://arxiv.org/abs/2603.07955
16. Jaewon Lee, Rational concordance of double twist knots, arXiv:2504.07636 (2025). DOI: 10.48550/arXiv.2504.07636, https://arxiv.org/abs/2504.07636
17. Aditya Dwivedi and Ramadevi Pichai, Double twist knots and lattice paths, arXiv:2605.02786 (2026). DOI: 10.48550/arXiv.2605.02786, https://arxiv.org/abs/2605.02786