A state enumeration of the foil knot

TL;DR

This paper introduces a novel enumeration of states in the foil knot, revealing a surprising connection to the lazy caterer's sequence and establishing a bijection with planar partitions.

Contribution

It presents a new state enumeration method for the foil knot and uncovers a link to the lazy caterer's sequence, providing a combinatorial interpretation.

Findings

Number of two-component states matches the lazy caterer's sequence.

Established a bijection between foil knot states and planar line partitions.

Revealed a surprising combinatorial connection in knot state enumeration.

Abstract

We split the crossings of the foil knot and enumerate the resulting states with a generating polynomial. Unexpectedly, the number of such states which consist of two components are given by the lazy caterer's sequence. This sequence describes the maximum number of planar regions that is obtained with a given number of straight lines. We then establish a bijection between this partition of the plane and the concerned foil splits sequence.