Knots, Braids and First Order Logic

TL;DR

This paper models knot equivalence using first-order logic by translating knots into a formal language with axioms, enabling logical analysis of knot isotopy through algebraic and logical methods.

Contribution

It introduces a novel first-order logical framework for representing and analyzing knot equivalence via braids, connecting topology with formal logic.

Findings

Knots can be represented as terms in a first-order language.

Knot equivalence corresponds to logical equivalence of terms.

The framework provides a new perspective on knot theory using formal logic.

Abstract

Determining when two knots are equivalent (more precisely isotopic) is a fundamental problem in topology. Here we formulate this problem in terms of Predicate Calculus, using the formulation of knots in terms of braids and some basic topological results. Concretely, Knot theory is formulated in terms of a language with signature $(\cdot,T,\equiv, 1,\sigma,\bar\sigma)$, with $\cdot$ a 2-function, $T$ a 1-function, $\equiv$ a 2-predicate and 1, $\sigma$ and $\bar\sigma$ constants. We describe a finite set of axioms making the language into a (first order) theory. We show that every knot can be represented by a term $b$ in 1, $\sigma$, $\bs$ and $T$, and knots represented by terms $b_1$ and $b_2$ are equivalent if and only if $b_1\equiv b_2$. Our formulation gives a rich class of problems in First Order Logic that are important in Mathematics.