The Alexander Polynomial of a Knotted Trivalent Graph

Paperlets Navigation Panel   Algebraic Knot Theory Algebraic Knot Theory - A Call for Action. The Kontsevich Integral for Knotted Trivalent Graphs. Knotted Trivalent Graphs are Finitely Presented. VS, TS and TG Algebras. The Alexander Polynomial of a Knotted Trivalent Graph. Some Questions About Trinions. Finite Type Invariants The Envelope of a Finite Type Invariant. The Envelope of The Alexander Polynomial. Some Quotients of A. Associators with Frozen Feet. The HOMFLY Braidor Algebra. Coloured Chord Diagrams. Khovanov Homology Mutation Invariance of Khovanov Homology. Khovanov Homology of Alternating Tangles. Other The Existence of the Exponential Function. The Many Faces of Alexander.

Statement. The Alexander polynomial of a knotted trivalent graph $\gamma$ is the Reidemeister torsion of the singular homology complex of the complement of $\gamma$, with local coefficients twisted using the Alexander duality pairing with $H_{1}(\gamma )$. Thus is it a numerical function on $H_{1}(\gamma )$; in particular, if $\gamma$ is a link, it is a numerical function in as many variables as the number of components of the link. In this case it is given by a Laurant polynomial.

I don't properly understand all the words in this statement, so I'm not sure if it is true or close to being true. Yet my understanding of the statement matches the limited understanding I have of the notions that appear in it, and I strongly feel that if true, the statement would be valuable, as it will lead to a proper extension of the Alexander polynomial to be an Algebraic Knot Theory. It may well be, of course, that a statement like that is well known (at least to Turaev).