Even-stranded Turk's head knots have the Levine–Tristram signature function of T(2,q), and Problems Q and R of Di Prisa–Şavk for fixed p
Year: 2026
Type: preprint
Abstract: For p even and q >= 3 with gcd(p,q) = 1 we show that the Levine–Tristram signature function of the Turk's head knot Th(p,q) coincides with that of the torus knot T(2,q), away from the finitely many roots of the two Alexander polynomials. The proof combines the Gambaudo–Ghys formula for signatures of closed braids with the factorisation of det(I - x·Burau(A_p)) proved in our earlier note (doi:10.5281/zenodo.23154788), Squier's invariant Hermitian form for the reduced Burau representation (with a uniqueness lemma), and a Krein-signature count. Consequence: for every fixed even p >= 4, the knots Th(p,q), q >= 3 odd with gcd(p,q) = 1, are linearly independent in the concordance group C and in the rational concordance group C_Q. In the reading of Problems Q and R of Di Prisa and Şavk (arXiv:2409.20106) in which p is fixed — the reading their wording and context suggest — this answers both problems. If p is allowed to vary as well, these Levine–Tristram invariants cannot settle the problems: neither the concordance invariants σ_ω (ω in the dense set where they are concordance invariants) nor the jump function distinguishes Th(p,q) from Th(p',q), since the two signature functions agree off finitely many points of the circle. The deposit contains the note, its LaTeX source, and the scripts and outputs of every check the note cites. STATUS: not refereed. This work was developed in collaboration with the AI system Claude Code (Anthropic): the choice of problem, the direction of the work and the computing hardware are due to the author (a sculptor); the formalisation, the algorithms and the proofs were developed by Claude Code (Anthropic) working with the author. Details are in the section on tool use at the end of the note. Licences: text CC BY 4.0; code GNU General Public License v3.0 or later; data CC BY 4.0.
Author: Stanggassinger, Werner Alois
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225786647