Knots, Braids and First Order 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.