The Khovanov width of twisted links and closed 3-braids

TL;DR

This paper investigates the Khovanov width, a measure derived from Khovanov homology, and demonstrates how to generate infinite link families with identical widths, providing explicit calculations for all closed 3-braids.

Contribution

It introduces methods to generate infinite families of links with the same Khovanov width and computes this width for all closed 3-braids.

Findings

Infinite families of links with identical Khovanov width can be generated.

Khovanov width is computed explicitly for all closed 3-braids.

Support of Khovanov homology lies on finite slope two lines.

Abstract

Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate infinite families of links with the same Khovanov width from link diagrams satisfying certain conditions. Consequently, we compute the Khovanov width for all closed 3-braids.