Rectangular Seifert circles and arcs system

TL;DR

This paper demonstrates how to deform oriented link diagrams into rectangular forms with bounds on the number of vertical segments, using isotopies that simplify Seifert circles and arcs.

Contribution

It introduces a method to transform any oriented link diagram into a rectangular diagram with explicit bounds based on crossings and Seifert circles, improving diagram simplification techniques.

Findings

Deformation into rectangular diagrams with at most c(D)+2s(D) vertical segments.

For connected diagrams, at most 2c(D)+2-w(D) vertical segments.

Seifert circles can be made into rectangles with two vertical and two horizontal segments.

Abstract

Rectangular diagrams of links are link diagrams in the plane ${\mathbb R}^2$ such that they are composed of vertical line segments and horizontal line segments and vertical segments go over horizontal segments at all crossings. P. R. Cromwell and I. A. Dynnikov showed that rectangular diagrams of links are useful for deciding whether a given link is split or not, and whether a given knot is trivial or not. We show in this paper that an oriented link diagram $D$ with $c(D)$ crossings and $s(D)$ Seifert circles can be deformed by an ambient isotopy of ${\mathbb R}^2$ into a rectangular diagram with at most $c(D) + 2 s(D)$ vertical segments, and that, if $D$ is connected, at most $2c(D)+2-w(D)$ vertical segments, where $w(D)$ is a certain non-negative integer. In order to obtain these results, we show that the system of Seifert circles and arcs substituting for crossings can be deformed by an ambient isotopy of ${\mathbb R}^2$ so that Seifert circles are rectangles composed of two vertical line segments and two horizontal line segments and arcs are vertical line segments, and that we can obtain a single circle from a connected link diagram by smoothing operations at the crossings regardless of orientation.