Transformations of lattice diagrams and their associated dotted diagrams

TL;DR

This paper refines the concept of reduced diagrams for lattice diagrams by introducing dotted diagrams, exploring their deformations and relations to transformations, building on previous work.

Contribution

It introduces admissible dotted diagrams for lattice diagrams and studies their deformations, refining prior results on diagram transformations.

Findings

Refined the notion of reduced diagrams with dotted diagrams.

Established relations between dotted diagram deformations and lattice diagram transformations.

Corrected and improved previous lemmas and theorems.

Abstract

We consider a graph called a lattice diagram, which is a graph in the $xy$-plane such that each edge is parallel to the $x$-axis or the $y$-axis. In [4], we investigated transformations of certain lattice diagrams, and we considered the reduced diagram that is obtained from deformations of a diagram associated with a lattice diagram. In this paper, we refine the notion of the reduced diagram by introducing the notion of a dotted diagram. A lattice diagram is presented by an admissible dotted diagram. We investigate deformations of dotted diagrams, and we investigate relation between deformations of admissible dotted diagrams and transformations of lattice diagrams, giving results that are refined and corrected versions of [4, Lemma 6.2, Theorem 6.3].