Virtual knots undetected by 1 and 2-strand bracket polynomials

TL;DR

This paper demonstrates the limitations of 1 and 2-strand bracket polynomials in detecting certain non-trivial virtual knots, and introduces new infinite families of such knots undetected by these invariants.

Contribution

The authors construct specific virtual knot diagrams and infinite families that are not detected by low-strand bracket polynomials, highlighting their limitations.

Findings

Kishino's knot is not detected by the bracket polynomial but distinguished by 3-strand invariants.

Constructed virtual knots are non-trivial yet undetected by 1 and 2-strand bracket polynomials.

Infinite families of virtual knots are shown to be undetectable by these polynomials.

Abstract

Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, $ K_D $ and $ K_m $, that are not not detected by the bracket polynomial or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, we note these virtual knot diagrams are trivial as flats.