Enumerating the states of the twist knot

TL;DR

This paper counts the state diagrams of twist knot shadows, linking them to plane partitions by circles, and introduces a bijection with binary words to understand their structure.

Contribution

It establishes a novel enumeration of twist knot states and connects this enumeration to geometric plane partitions through a binary word bijection.

Findings

Number of twist knot states equals the maximum regions divided by circles.

A bijection between state enumeration and plane partitions is constructed.

The enumeration aligns with known geometric partition counts.

Abstract

We enumerate the state diagrams of the twist knot shadow which consist of the disjoint union of two trivial knots. The result coincides with the maximal number of regions into which the plane is divided by a given number of circles. We then establish a bijection between the state enumeration and this particular partition of the plane by means of binary words.