From the lambda calculus and (some of) its “classic” graphical representations developed by Zeilberger, Hasegawa introduced a new lambda calculus, the “braided lambda calculus”, that keeps track of some geometrical interactions appearing naturally in Zeilberger’s work. From the braided lambda calculus, he developed a sound and complete axiomatization of various categorical settings using an SKI-style system of combinators. Here, we will expand upon this axiomatization by dealing with trace operators (and therefore trace combinators). Surprisingly, from this axiomatization, surprisingly, this extension yields — almost “for free” — the more categorically demanding duality operators, echoing Joyal, Street, and Verity’s famous Int construction, but in a purely combinatorial framework. Finally, using some well-known interactions of those categorical settings with geometry, we will get a framework in which certain lambda terms can interact with low-dimensional topological objects like braids and knots, opening the way to, maybe, a new kind of computationally anchored geometric invariant.
