Combinatorial Maps with Normalized Knot

TL;DR

This paper introduces the concept of normalized knots in combinatorial maps, demonstrating that normalization simplifies corner numeration and cycle identification without limiting generality, applicable to edge structuring knots as well.

Contribution

It presents a novel normalization method for combinatorial map knots that enhances clarity and consistency in corner and cycle representation.

Findings

Normalization does not affect the generality of combinatorial maps.

Normalized knots lead to more concise corner numeration.

Both map and edge knots can be normalized mutually.

Abstract

We consider combinatorial maps with fixed combinatorial knot numbered with augmenting numeration called normalized knot. We show that knot's normalization doesn't affect combinatorial map what concerns its generality. Knot's normalization leads to more concise numeration of corners in maps, e.g., odd or even corners allow easy to follow distinguished cycles in map caused by the fixation of the knot. Knot's normalization may be applied to edge structuring knot too. If both are normalized then one is fully and other partially normalized mutually.