# A companion to "Knot invariants and M-theory I'' [arXiv:1608.05128]: proofs and derivations

## TL;DR

This paper develops two M-theory models and a related four-dimensional gauge theory framework to study knot invariants, providing detailed derivations and insights into supersymmetry and topological twists, complementing prior work.

## Contribution

It offers a detailed mathematical derivation and an alternative formulation of knot invariants within M-theory and four-dimensional gauge theories, expanding understanding of their physical and mathematical structures.

## Key findings

- Construction of two related M-theory models for knot invariants
- Derivation of a four-dimensional gauge theory hosting knots
- Insights into supersymmetry and topological twists

## Abstract

We construct two distinct yet related M-theory models that provide suitable frameworks for the study of knot invariants. We then focus on the four-dimensional gauge theory that follows from appropriately compactifying one of these M-theory models. We show that this theory has indeed all required properties to host knots. Our analysis provides a unifying picture of the various recent works that attempt an understanding of knot invariants using techniques of four-dimensional physics. This is a companion paper to arXiv:1608.05128, covering all but section 3.3. It presents a detailed mathematical derivation of the main results there, as well as additional material. Among the new insights, those related to supersymmetry and the topological twist are highlighted. This paper offers an alternative, complementary formulation of the contents in~\cite{Dasgupta:2016rhc}, but is self-contained and can be read independently.