Piecewise-linear pseudodiagrams

TL;DR

This paper studies piecewise-linear pseudodiagrams, focusing on how their resolutions differ from smooth cases in three-dimensional space, introducing the forcing number and classifying shadows with unique resolution sets.

Contribution

It introduces the concept of the forcing number for PL pseudodiagrams and classifies shadows based on their weighted resolution sets compared to smooth pseudodiagrams.

Findings

Certain PL pseudodiagram resolutions do not exist in three-space.

The weighted resolution set can differ between PL and smooth cases.

Classification of shadows with differing resolution sets.

Abstract

There are 2^n possible resolutions of a smooth pseudodiagram with n precrossings. If we consider piecewise-linear (PL) pseudodiagrams and resolutions that themselves are PL, certain resolutions of the pseudodiagram may not exist in three-space. We investigate this situation and its impact on the weighted resolution set of PL pseudodiagrams as well as introduce a concept specific to PL pseudodiagrams, the forcing number. Our main result classifies the PL shadows whose weighted resolution sets differ from the weighted resolution set that would exist in the smooth case.