A remarkable 20-crossing tangle

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 $\times$ 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.