Banded surfaces, banded links, band indices and genera of links

TL;DR

This paper introduces the concepts of banded surfaces, band indices, and their relation to link genera, providing bounds and calculations for specific classes like pretzel knots.

Contribution

It defines band indices for links and explores their relationship with link genus, offering bounds and explicit calculations for pretzel knots.

Findings

Banded links of small band index are characterized.

Upper bounds for band indices are established using braid and Seifert surfaces.

Band indices of pretzel knots are explicitly calculated.

Abstract

Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of bands required to present the link as boundaries of an unknotted (flat) banded surface. Banded links of small (flat, respectively) band index are considered here. Some upper bounds are provided for these band indices of a link using braid representatives and canonical Seifert surfaces of the link. The relation between the band indices and genera of links is studied and the band indices of pretzel knots are calculated.