The minimum crossing number of essential tangles

TL;DR

This paper establishes the exact lower bounds for the crossing number of essential tangles with multiple strings and loops, and characterizes minimal crossing configurations, advancing understanding in knot theory.

Contribution

It provides the first precise lower bounds for crossing numbers of essential tangles and characterizes minimal cases for various configurations.

Findings

Sharp lower bounds for crossing numbers of essential tangles.

Characterization of minimal crossing number tangles with unknotted and knotted strings.

Complete classification of minimal crossing configurations for certain tangle types.

Abstract

In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given number of components, both when the tangle has knotted strings or only unknotted strings.