Knot projections with reductivity two

TL;DR

This paper classifies knot projections with reductivity two across four different definitions, introduces new reductivities, and provides bounds and properties, advancing understanding of knot simplification processes.

Contribution

It determines all knot projections with reductivity two for four definitions, including three new types, and explores their properties and relationships.

Findings

Identified all knot projections with reductivity two for four definitions.

Introduced three new reductivities related to knot projections.

Provided lower bounds and properties for each reductivity type.

Abstract

Reductivity of knot projections refers to the minimum number of splices of double points needed to obtain reducible knot projections. Considering the type and method of splicing (Seifert type splice or non-Seifert type splice, recursively or simultaneously), we can obtain four reductivities containing Shimizu's reductivity, three of which are new. In this paper, we determine knot projections with reductivity two for all four of the definitions. We also provide easily calculated lower bounds for some reductivities. Further, we detail properties of each reductivity, and describe relationships among the four reductivities with examples.