The unknotting number of the knot $10_6$: deciding between $2$ and $3$

The unknotting number $u(K)$ — the minimum number of crossing changes needed to turn a knot $K$ into the unknot — is not known for the knot $10_6$ in the standard Rolfsen knot table: the signature bound gives $u(10_6)\ge 2$, and an explicit three-crossing unknotting sequence in its minimal diagram gives $u(10_6)\le 3$. Brittenham and Hermiller (arXiv:2601.18757, 2026), who produced the smallest known failures of additivity of unknotting number under connected sum ($u(4_1\#9_{10})\le 3<4$ and $u(5_1\#8_2)\le 3<4$), also constructed a 15-crossing diagram of $3_1\#10_6$ that three crossing changes unknot. Determining $u(10_6)$ therefore settles a sharply posed dichotomy: if $u(10_6)=3$, then $u(3_1\#10_6)\le 3<1+3$ is a genuine counterexample to additivity of unknotting number with a trefoil summand, and dozens of further small knots acquire 'symbionts' via Gordian adjacency; if $u(10_6)=2$, then $10_6$ becomes the smallest-crossing counterexample to the Bernhard–Jablan Unknotting Conjecture. No determination of the value appears in the literature, so the problem remains open.

Background

A crossing change is the local operation on a knot diagram that switches which of the two strands passes over at a single crossing, and the unknotting number $u(K)$ of a knot $K$ is the minimum number of crossing changes, over all diagrams of $K$, required to obtain the unknot. The connected sum $K_1\#K_2$ of two knots is formed by deleting a short arc from each of two disjoint diagrams and splicing the endpoints together; knots that are not themselves connected sums are called prime, and prime knots are tabulated by crossing number, with $3_1$ (the trefoil), $4_1$ (the figure-eight knot), $5_1$, and so on denoting the standard Rolfsen-table entries. Whether unknotting number is additive under connected sum, $u(K_1\#K_2)=u(K_1)+u(K_2)$, was a classical open question for decades. The knot signature $\sigma(K)$ — an integer computable from a Seifert surface of $K$ (an oriented surface in 3-space whose boundary is the knot) — provides the standard lower bound $u(K)\ge|\sigma(K)|/2$ and is additive under connected sum (with mirroring signs mattering: for the mirror image $\overline{K}$ one has $\sigma(\overline{K})=-\sigma(K)$, while $u(\overline{K})=u(K)$).

Two named conjectures frame the problem. First, the additivity question was answered negatively by Brittenham and Hermiller in 2025, who showed $u(7_1\#\overline{7_1})\le 5<6=u(7_1)+u(\overline{7_1})$, where $7_1$ is the $(2,7)$-torus knot (the knot wrapping twice around one direction and seven times around the other of a standard torus). Their 2026 follow-up proved the far smaller failures $u(4_1\#9_{10})\le 3<4=u(4_1)+u(9_{10})$ and $u(5_1\#8_2)\le 3<4=u(5_1)+u(8_2)$, and coined the term symbionts: knots $K$ and $K'$ are symbionts if $u(K\#K')<u(K)+u(K')$. They also use Gordian adjacency — $K_1$ is Gordian adjacent to $K_2$ if $K_1$ occurs in some minimal-length unknotting sequence for $K_2$ — to propagate symbiosis: if $K_2$ is a symbiont for $K_1$ and $K_1$ is Gordian adjacent to $K$, then $K_2$ is a symbiont for $K$.

Second, the Bernhard–Jablan Unknotting Conjecture asserted that every knot $K$ has a minimum-crossing-number diagram in which some single crossing change produces a knot $K'$ with $u(K')=u(K)-1$, which would make $u$ recursively computable from minimal diagrams alone. Brittenham and Hermiller disproved this, and the disproof isolates the recursively computed quantity $u_{BJ}(K)$ — the Bernhard–Jablan unknotting number, an effectively computable upper bound for $u(K)$ obtained by taking, over all crossing changes in all minimal diagrams of $K$, one plus the smallest value of $u_{BJ}$ of the resulting knot. Any knot with $u(K)<u_{BJ}(K)$ is a counterexample to the conjecture; the known counterexamples have 12 or more crossings.

The present problem concerns $10_6$. Its unknotting number was reported as $3$ in tables from the 1990s onward, but Brittenham and Hermiller document that this traces to a typo in a 1996 reference survey, corrected in its author's online errata to '$2$ or $3$', a correction incorporated into the KnotInfo database. The bounds are tight but not coincident: $|\sigma(10_6)|=4$ gives $u(10_6)\ge 2$, and in the unique (up to flypes — a local reshuffling move on diagrams) minimal 10-crossing diagram there are 35 different triples of crossing changes that produce the unknot, so $u(10_6)\le 3$. The same 2026 paper constructs an explicit 15-crossing diagram of $3_1\#10_6$, with mirroring chosen so that $|\sigma(3_1\#10_6)|=2$, together with a three-crossing unknotting sequence, giving $u(3_1\#10_6)\le 3$ (the paper supplies SnapPy-verifiable DT codes, i.e. Dowker–Thistlethwaite encodings of the diagrams). This sets up the dichotomy: the value of $u(10_6)$ alone decides between two different named open outcomes.

Problem Statement

Determine the exact value of the unknotting number of the knot $10_6$ (Rolfsen/KnotInfo table convention). It is known that $2\le u(10_6)\le 3$: the knot signature satisfies $|\sigma(10_6)|=4$, giving $u(10_6)\ge 2$, and an explicit three-crossing unknotting sequence in the minimal diagram gives $u(10_6)\le 3$. The value is therefore one of $2$ or $3$, and the two possible determinations, each of which settles the source paper's dichotomy, are:

1. $u(10_6)=2$: exhibit a diagram of $10_6$ and two crossing changes whose result is the unknot (together with verification that the starting diagram indeed represents $10_6$). Since $u_{BJ}(10_6)=3$, this would simultaneously make $10_6$ the smallest-crossing counterexample to the Bernhard–Jablan Unknotting Conjecture.

2. $u(10_6)=3$: prove a lower bound $u(10_6)\ge 3$, which combined with the known upper bound gives equality. This would simultaneously establish, via Brittenham and Hermiller's Theorem 1.3, that the connected sum $3_1\#10_6$ (with mirroring chosen so $|\sigma(3_1\#10_6)|=2$) satisfies $u(3_1\#10_6)\le 3<1+3=u(3_1)+u(10_6)$, a genuine counterexample to additivity of unknotting number with a trefoil summand, and would give symbionts to the 38 remaining at-most-10-crossing knots of unknotting number 2 that are Gordian adjacent to $3_1$.

Either answer is a single definitive determination; the unknotting number is invariant under mirroring, so no mirror convention enters the value itself. Progress claims that only improve one bound without closing the gap (e.g., new upper bounds of 3 or lower bounds of 2) do not answer the question.

References

1. 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
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
3. Mark Brittenham and Susan Hermiller, "A counterexample to the Bernhard–Jablan unknotting conjecture", arXiv:1705.05985 (2017), DOI 10.48550/arxiv.1705.05985, https://arxiv.org/abs/1705.05985
4. 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
5. Tye Lidman, "The unknotting number of $11n102$ is 2", arXiv:2606.12431 (2026), DOI 10.48550/arxiv.2606.12431, https://arxiv.org/abs/2606.12431
6. Michal Jablonowski, "Integer Knot Invariants: Inequalities, Computations, and Open Problems", arXiv:2605.22652 (2026), DOI 10.48550/arxiv.2605.22652, https://arxiv.org/abs/2605.22652
7. 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
8. 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
9. Martin G. Scharlemann, "Unknotting number one knots are prime", Inventiones Mathematicae 82 (1985), 37–55, DOI 10.1007/BF01394778, https://doi.org/10.1007/BF01394778