Length conjecture for twist knots
For every twist knot the all-order asymptotic prediction known as the length conjecture holds, refining the volume conjecture by incorporating insertions of additional links in the skein module of the knot complement. The supplied source does not state the conjecture's explicit asymptotic formula or define the associated length and link-insertion data.
References
Primary source
Additional references
Progress summary
A new paper verifies the conjecture’s first-order prediction for infinitely many twist knots, but the full prediction remains open.
The length conjecture is a refinement of the volume conjecture for twist knots. The broader conjecture concerns all orders, whereas the reported result establishes only its first-order case for an infinite family.
September 2026 first-order result
A September 2026 arXiv paper claims a first-order proof for an infinite family of twist knots, providing new evidence for the length conjecture. It also proposes a related Dehn-filling naturality conjecture. The claimed result does not settle the all-order conjecture and has not been independently verified.
Current status (as of September 2026): first-order validity is claimed for an infinite family of twist knots, while the all-order length conjecture remains open and the claim is unverified.
Sources
- arxiv.org
- arxiv.org
- numdam.org
- normalesup.org
- semanticscholar.org
- inspirehep.net
- cnrs.hal.science
- researchgate.net
- deepmind.google
- ar5iv.labs.arxiv.org
- export.arxiv.org
- arxiv.org
- arxiv.org
- mathstodon.xyz
- quantamagazine.org
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
- quantamagazine.org
- scientificamerican.com
- quantamagazine.org
- x.com
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