Evolution properties of the knot's defect

TL;DR

This paper investigates the invariance and linear change of the defect in differential expansions of colored HOMFLY-PT polynomials under antiparallel and parallel evolutions, respectively, revealing fundamental properties of knot invariants.

Contribution

It proposes a conjecture describing how the defect behaves under different types of evolutions, connecting the properties of R-matrices and braid modifications in knot theory.

Findings

Defect remains invariant under antiparallel evolution.

Defect changes linearly with parallel evolution.

Provides a new perspective on the behavior of knot invariants under braid modifications.

Abstract

The defect of differential (cyclotomic) expansion for colored HOMFLY-PT polynomials is conjectured to be invariant under any antiparallel evolution and change linearly with the evolution in any parallel direction. In other words, each ${\cal R}$-matrix can be substituted by an entire 2-strand braid in two different ways: the defect remains intact when the braid is antiparallel and changes by half of the added length when the braid is parallel.