I will provide a test question. Below tell me is the question Solvable currently or not: Produce an algorithm that takes as input a diagram of a knot and returns True if the unknotting number of the knot represented by that diagram is 1 and False otherwise. For a knot with up to 100 crossings, the algorithm must complete in under an hour when run on a typical laptop. Provide your solution as a self-contained Python script. Accept 1-indexed planar crossing diagrams as the input format. Example input: [[4,2,5,1],[10,6,11,5],[8,3,9,4],[2,9,3,10],[11,16,12,17],[7,15,8,14],[15,7,16,6],[13,20,14,21],[17,22,18,1],[21,18,22,19],[19,12,20,13]]

Sonar Pro

Vote: No

The question asks for an algorithm that, given a knot diagram with up to 100 crossings, determines in under an hour on a typical laptop whether the unknotting number is exactly 1. This is not solvable currently. Determining if the unknotting number is 1 is an open problem in knot theory, even for this restricted case, as stated explicitly in search result [3]: 'It is not known at present whether an algorithm exists for... this special case.' Result [5] notes that unknotting number is not known to be algorithmically computable, with even an 8-crossing knot having unknown unknotting number. Recent advances use reinforcement learning to find upper bounds by discovering unknotting sequences (e.g., for up to 200 crossings in [2] and [6]), but this only provides upper bounds, not exact values. To confirm exactly 1, one must prove no single crossing change unknots it, which requires exhaustive search over all possibilities—exponentially many for 100 crossings, infeasible in under an hour. No self-contained Python script solving this exists, as it would resolve a major open problem in low-dimensional topology.

GPT-5.4

Vote: No

This is not currently solvable as stated. The key issue is not mere decidability in principle, but the requirement for a practical, self-contained algorithm that takes an arbitrary knot diagram with up to 100 crossings and correctly decides whether the unknotting number is exactly 1 within an hour on a typical laptop. There is no known general-purpose algorithm in current knot theory software/practice that can reliably do this for all such inputs. Unknotting number is notoriously hard to compute. Even determining unknotting number 1 is difficult in general, and standard invariants usually give only obstructions or partial confirmations, not a complete decision procedure fast enough for all 100-crossing diagrams. A brute-force approach of trying all crossing changes and then testing for the unknot is insufficient from a diagram alone, because a knot with unknotting number 1 may require a crossing change not present in the given diagram unless one first searches through Reidemeister-equivalent diagrams. That search space is enormous and not known to be controllable within the stated runtime bound. So while one can build heuristic or partial algorithms, the requested guaranteed correct under-an-hour solver for arbitrary inputs up to 100 crossings is beyond current practical/known methods.

Grok 4.1 Fast

Vote: No

Computing the unknotting number for a knot diagram with up to 100 crossings is currently infeasible in under an hour on a laptop, especially in a self-contained Python script. Unknot recognition alone is decidable but practically challenging for large diagrams; Reidemeister moves explode combinatorially, and no efficient pure-Python implementation exists for 100 crossings. Recent theoretical polynomial-time algorithms (e.g., Lackenby's) are not yet practical or implemented. To check unknotting number exactly 1 requires: (1) confirming the knot is non-trivial, (2) testing all ~100 single crossing changes, each needing full unknot recognition on the modified diagram. Each check would exceed seconds to minutes even with optimized tools like SnapPy/Regina, totaling far over an hour. No knot theory library is standard Python; self-contained means no external deps like SnapPy. Invariants (Jones, Alexander) are computable but incomplete for unknot detection. Tables exist only up to ~16 crossings. Thus, no such algorithm exists today.

Claude Opus 4.6

Vote: No

Determining the unknotting number of a knot is a notoriously difficult problem in knot theory. While there are known lower bounds (from signatures, Rasmussen's s-invariant, etc.) and upper bounds (by trying crossing changes), computing the exact unknotting number is not known to be efficiently decidable in general. For the specific case of unknotting number 1, there has been significant theoretical progress. Notably, work by Lackenby and others provides approaches, and the result that unknotting number 1 knots are prime combined with checking genus via normal surface theory gives theoretical algorithms. However, the practical constraint here is extremely demanding: the algorithm must handle knots with up to 100 crossings and complete within an hour on a typical laptop. The brute-force approach of trying all possible crossing changes (there are O(n) crossings to try, and for each you'd need to determine if the result is the unknot) is feasible in principle for the 'upper bound' direction — you can try each crossing change and check if the result is the unknot (unknot recognition is decidable and practically feasible). This would confirm unknotting number ≤ 1. But the hard part is proving that the unknotting number is NOT 1 when it isn't. For proving u(K) ≥ 2, you need invariant computations. The signature gives a lower bound, and Rasmussen's s-invariant (from Khovanov homology) gives another. However, computing Khovanov homology for 100-crossing knots within an hour is not feasible — Khovanov homology computation is exponential in the number of crossings, and 100 crossings would be far beyond current practical limits. Alternatively, one could use the knot signature (computable from the Seifert matrix) which is polynomial-time, but the signature alone doesn't always detect unknotting number > 1. There exist knots with signature 0 but unknotting number > 1. Lackenby's recent work (2021) showed unknot recognition is in co-NP, but this doesn't directly give a practical polynomial-time algorithm for unknotting number 1. There is no known practical algorithm that can definitively determine unknotting number 1 for arbitrary knots up to 100 crossings within the time constraint. In summary, while the 'yes' direction (u(K) = 1) can be checked by brute force over crossing changes + unknot recognition, the 'no' direction (proving u(K) ≠ 1) lacks a practical algorithm that works reliably for all knots up to 100 crossings within an hour. Therefore, this problem is not currently solvable under the stated constraints.

Gemini 3.1 Pro

Vote: No

The question asks for an algorithm that can determine if a given knot has an unknotting number of exactly 1. In knot theory, the unknotting number is the minimum number of crossing changes required to transform a knot into the unknot. Currently, it is a major open problem in topology whether the unknotting number of a knot is algorithmically computable. More specifically, it is not even known to be mathematically decidable whether an arbitrary knot has an unknotting number of 1. Because the decidability of this property remains an unsolved problem, no such algorithm exists today. Furthermore, even if it were theoretically decidable, doing so for diagrams with up to 100 crossings in under an hour would be practically impossible with current techniques, as the problem involves an incredibly vast search space of possible isotopic diagrams. Therefore, the task is currently unsolvable.

Kimi K2.5

Vote: No

Determining whether a knot has unknotting number exactly 1 is algorithmically intractable under the given constraints for two key reasons. First, verifying that the unknotting number is nonzero requires solving the unknot recognition problem (distinguishing any nontrivial knot from the unknot). While unknot recognition is known to be in NP ∩ co-NP, it is not known to be in P, and the best-known algorithms have exponential worst-case complexity. For arbitrary 100-crossing diagrams, current state-of-the-art methods (using normal surface theory or Heegaard Floer homology) cannot guarantee completion within one hour on a typical laptop for all instances. Second, and more critically, verifying that the unknotting number is at most 1 requires finding a certificate across all possible diagrams of the knot, not just the input diagram. The unknotting number is defined as the minimum over all diagrams and all crossing sets. To prove u(K) ≤ 1, one must find some diagram (reachable via Reidemeister moves) where a single crossing change unknots the knot. However, it is known that transforming a diagram into a simplifying form may require exponentially many Reidemeister moves. Consequently, the search space is vast and unstructured. Furthermore, recent complexity results indicate that deciding whether u(K) ≤ 1 is NP-hard. Unless P = NP, no polynomial-time algorithm exists, and exhaustive search over potential unknotting sets becomes prohibitive for 100 crossings. Therefore, no currently known algorithm can reliably meet the specified performance requirements.

Final Summary

The AI Roundtable reached a swift and unanimous verdict: current mathematics and computing power are simply not up to the task of deciding unknotting numbers for 100-crossing diagrams in an hour. While Claude Opus 4.6 and Kimi K2.5 highlighted the theoretical bottlenecks in proving a knot *isn't* unknotting number 1, Gemini 3.1 Pro pointed out the even deeper issue that the problem's mathematical decidability itself remains an open question in topology.

No agreement after 1 rounds of discussion. Final vote: No: 6

Strongest Arguments

• No: Determining whether a knot has unknotting number 1 is an open problem in topology, and even if decidable, the computational cost of verifying the result across the vast search space of 100-crossing diagrams exceeds the one-hour laptop constraint.