A remarkable 20-crossing tangle
Shalom Eliahou (LMPA), Jean Fromentin (LMPA)

TL;DR
This paper constructs knots with a specific number of crossings whose Jones polynomial exhibits particular modular properties, revealing new insights into knot invariants and their detectability.
Contribution
It introduces a 20-crossing tangle that produces knots with Jones polynomials equal to 1 modulo 2 r, advancing understanding of knot invariants and their limitations.
Findings
Constructed knots with (20 × 2 r - 1 + 1) crossings
Jones polynomial equals 1 mod 2 r for these knots
Identified a 20-crossing tangle undetectable by Kauffman bracket polynomial mod 2
Abstract
For any positive integer r, we exhibit a knot Kr with (20 2 r--1 + 1) crossings whose Jones polynomial V (Kr) is equal to 1 mod-ulo 2 r. Our construction rests on a certain 20-crossing tangle T 20 which is undetectable by the Kauffman bracket polynomial pair mod 2.
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