Abstract:We develop a finite-type framework for studying the Jones polynomial and its ability to distinguish the unknot. The main difficulty is to propagate the vanishing of its finite-type coefficient layers from the natural upper degree bound down to the first potentially informative orders. To overcome this, we introduce a local clasped-twist construction whose diagrammatic reductions remain uniformly controlled even when the twist is made arbitrarily long. This separates the role of the twist length from the degree bound and permits a descending vanishing argument. A single construction transfers high-order finite-type vanishing to a long residual twist, where local smoothing relations and an anchor reproducing the original knot force vanishing at all relevant lower orders. Two constructions placed in disjoint regions then generate a controlled two-crossing family. On this family, the resulting low-order relations impose a component count after successive smoothings that contradicts the topology of a suitably chosen pair of crossings. As a consequence, every nontrivial knot has a nonzero Jones finite-type coefficient at an order bounded above by three times its crossing number. In particular, a knot whose Jones polynomial is equal to that of the unknot must itself be the unknot.
From: Avishy Carmi [view email]
[v1]
Sun, 21 Jun 2026 09:56:10 UTC (3,960 KB)
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