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

TL;DR

This work shows that braided satellites preserve HOMFLY coefficient polynomials up to a prescribed $z$-degree across infinite families of distinct knots. By linking coefficient polynomials to finite type invariants and leveraging $n$-equivalence, the authors construct infinitely many mutually distinct hyperbolic knots $\{K_i\}$ that share the same satellite coefficient data with a fixed pattern class, and extend the result to $(q,1)$-cables. The method relies on patterns in solid tori, the FDTC of braids, and controlled perturbations inside the lower central series to generate infinitely many non-equivalent knots while maintaining coefficient polynomial equality. An analogous argument applies to the Dubrovnik version of the Kauffman polynomial, though the arc index constraint makes universal infinite families harder to realize in that setting. Overall, the paper elucidates the robustness of coefficient polynomials under satellite operations and delineates limits in distinguishing knots via these polynomial invariants.

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 $w$.

Theorems & Definitions (16)

• Definition 1
• Theorem 2
• Corollary 1
• Proposition 1
• proof
• Theorem 3
• Corollary 2
• Corollary 3
• proof
• Proposition 2