Symmetric quotients of knot groups and a filtration of the Gordian graph

TL;DR

This paper introduces a new filtration of the Gordian graph based on knot group quotients to study knot properties and verifies the Meridional Rank Conjecture for a specific family of complex knots.

Contribution

It defines a novel metric filtration of the Gordian graph using symmetric group quotients and confirms the Meridional Rank Conjecture for certain high-bridge-number knots.

Findings

Established a filtration of the Gordian graph with dense subgraphs.

Verified the Meridional Rank Conjecture for knots with unknotting number one.

Connected group surjections to symmetric groups with transpositions to knot properties.

Abstract

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The n-th subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.