Computational Methods in Kirby Calculus.

Abstract

Kirby Calculus gives us the tools to analyze 4-dimensional manifolds based off of their handlebody decomposition. The manifolds can be described in terms of their handlebody decomposition via Kirby diagrams. One can transform one manifold into another by performing Reidemeister moves, handle creation, handle annihilation, and handle slides to adjust the diagram. We created a python package called MetaKnight (Manifolds Encoded Through the Architecture of Knots and Numbers In the Geometry of Handlebody Theory), which the user can use to create Kirby diagrams, and perform all types of moves on them. The first part of this paper explains Kirby Calculus, Handlebody Theory, as well as some Morse Theory and Knot Theory. The second part of this paper goes through the MetaKnight python package and explains the various functions and methods available for public use.