# Braid group and leveling of a knot

## TL;DR

This paper introduces a new invariant called the (1, 1)-length for knots in genus-1 1-bridge positions, showing it equals the level number, and provides braid descriptions for 2-bridge knots to estimate their level numbers.

## Contribution

It establishes the equivalence between the (1, 1)-length and the level number, and offers braid descriptions for all 2-bridge knots to bound their level numbers.

## Key findings

- (1, 1)-length equals the level number for knots in genus-1 1-bridge position.
- Braid descriptions for all 2-bridge knots are provided.
- The (-2, 3, 7)-pretzel knot has level number two.

## Abstract

Any knot $K$ in genus-$1$ $1$-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the $(1, 1)$-length. We show that the $(1, 1)$-length equals the level number. We then find braid descriptions for $(1,1)$-positions of all $2$-bridge knots providing upper bounds for their level numbers, and also show that the $(-2, 3, 7)$-pretzel knot has level number two.