Meta-monoids, meta-bicrossed products, and the Alexander polynomial (Q2859565)

This paper is based on lectures given by the first author, and is dependent only on a basic knowledge of sets, monoids and groups, introducing the other algebraic structures required. At the time of writing, further details of the lectures are available on the first author's website.NEWLINENEWLINEThe most basic object of concern in the paper is a list of elements of a set \(X\), labelled by elements of a finite set. Of interest are collections of such lists, together with several operations satisfying certain axioms. For a `meta-monoid', these operations can be loosely interpreted as relabelling, multiplication, deletion and creation of list items, and combining lists. This can be seen as a generalisation of a `monoid computer', in which \(X\) is a monoid and the operations actually have these interpretations.NEWLINENEWLINEA `meta-bicrossed product' has a similar construction. In this case the lists are labelled by elements of a pair of finite sets, and each element of \(X\), in some sense, has two `parts'. In addition, there are operations that roughly equate to multiplying the first parts, multiplying the second parts, and swapping over the parts.NEWLINENEWLINEUsing this framework, the authors construct an invariant of oriented tangles. They define a particular meta-bicrossed product \(\beta\). Next they define an element of \(\beta\) from any oriented tangle diagram, and finally the operations in \(\beta\) are used to produce a set of polynomials. Included in the paper is the Mathematica code to implement this procedure. Using this, the authors verify that the polynomials produced are a tangle invariant. Based on experimental evidence, they conjecture that these polynomials contain the Alexander polynomial, giving it precisely in the case of a single-string tangle.