Knots whose braided satellite have the same HOMFLY polynomial up to given $z$-degrees

TL;DR

This paper constructs infinitely many hyperbolic knots sharing the same HOMFLY polynomial up to certain degrees when considering their satellite knots with specific braided patterns, revealing limitations of polynomial invariants in knot distinction.

Contribution

It introduces a method to generate infinitely many distinct hyperbolic knots with identical HOMFLY polynomials up to specified degrees for a broad class of satellite knots.

Findings

Constructed infinite families of knots with identical HOMFLY polynomials up to given degrees.

Showed the limitations of HOMFLY polynomial in distinguishing certain satellite knots.

Demonstrated the existence of hyperbolic knots sharing polynomial invariants with others.

Abstract

For a given knot $K$ and $w>0$, we construct infinitely many mutually distinct hyperbolic knots $\{K_i\}$ such that the $P$-satellites of $K$ and $K_i$ have the same HOMFLY polynomial up to given $z$-degrees, for all braided patterns $P$ with winding number less than or equal to $w$.