A Diagrammatic Multivariate Alexander Invariant of Tangle

Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. In this dissertation, I will present my multivariate version of Bigelow's algorithm for the Alexander polynomial. The advantage to my algorithm is that it generalizes easily to a multivariate tangle invariant. I will also present preliminary results on the connection to Jana Archibald's tangle invariant and conclude with ideas for future research.