I found this here:
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 Laur[e]nt polynomial.
Can anybody explain this to me in detail?
- What is a knotted trivalent graph?
- What is the Reidemeister torsion of the singular homology complex of the complement of $\gamma$, with local coefficients twisted using the Alexander duality pairing with $H_1(\gamma)$?
- What does it mean that $H_1(\gamma)$ is given by a Laurent-Polynomial, if $\gamma$ is a link?
I checked already the Alexander Duality Pairing at Wikipedia, but it wasn't of great help...
A knotted trivalent graph is a (tame) embedding of a (finite) 1-dimensional CW complex (i.e., a graph) into $S^3$, up to isotopy, such that each vertex has three incident edges. These are also known as spatial graphs. "3-valent", "cubic", and "3-regular" mean the same thing. I don't think 3-valent is necessary for this invariant, and instead this reflect's Dror Bar-Natan's interests.
Let's get back to Reidemester torsion later, but Alexander duality says $H^1(\gamma)\cong H_1(S^3-\gamma)$, so there is a pairing between $H_1(\gamma)$ and $H_1(S^3-\gamma)$. (It can be thought of as computing linking number between a loop in $\gamma$ and a loop in $S^3-\gamma$.) This is only important for pinning down what the variables are in the Laurent polynomials.
$H_1(\gamma)$ is the homology of the graph. Recall (Hatcher, 1.A) that a graph is homotopy equivalent to a wedge sum of circles, one circle per edge outside a maximal spanning tree, so $H_1(\gamma)$ is $\mathbb{Z}^n$ for some $n$. Equivalently, if $t_1,\dots,t_n$ are generators of the various $\mathbb{Z}$'s in the direct sum, $H_1(\gamma)=\mathbb{Z}[t_1^{\pm 1},\dots,t_n^{\pm 1}]$, which is a ring of Laurent polynomials. There is an arbitrary choice here in the orientations of these generators.
Reidemeister torsion is an invariant of acyclic chain complexes (that is, chain complexes whose homology is zero), up to some factor that can be controlled by making some choices. My understanding is, given $S^3-N(\gamma)$, one can take the singular chain complex for the universal cover, twist it by tensoring with a representation of the fundamental group ($H_1(\gamma)$ in this case), and then compute the Reidemeister torsion, the details of which I won't discuss here, but here are some references that discuss it (Infinite cyclic coverings only indirectly, and I must admit I have not fully read these):
The resulting torsion is representable as an element of $H_1(\gamma)$ and can be written as a Laurent polynomial.
There is a version of this without invoking the words "Reidemeister torsion," and I believe it is supposed to be related, but I am not yet sure how to compute the torsion out of it if it is. Let $X=S^3-\gamma$, and let $\hat{X}$ be the maximal abelian cover of $X$. That is, $\pi_1(\hat{X})$ is the commutator subgroup $\pi_1(X)^{(1)}$ of $\pi_1(X)$. Then, the group of deck transformations of $\hat{X}$ is the abelianization of $\pi_1(X)$, namely $H_1(X)\cong H_1(\gamma)\cong\mathbb{Z}[t_1^{\pm 1},\dots,t_n^{\pm 1}]=\Lambda$. We can think of $H_i(\hat{X})$ as a $\Lambda$-module in the following way: define $t_i\cdot [\alpha]=[t_i\circ\alpha]$ (that is, by translation of cycles using the deck transformation). Unfortunately, the ring $\Lambda$ is not a PID so the structure theorem does not apply, but one can compute the GCD of the minors of a presentation matrix like in the usual Alexander polynomial. This is in Rolfsen's Knots and Links for links (p190), and covered for graphs and higher-dimensional complexes in Kinoshita, Shin’ishi, Alexander polynomials as isotopy invariants. I, Osaka Math. J. 10, 263-271 (1958). ZBL0119.38801.
A downside to an Alexander polynomial invariant is that if two graphs have homeomorphic complements of regular neighborhoods of the graphs, they will have the same invariants. For example, the wedge sum of two circles, the theta graph, the unlinked handcuffs, and the linked handcuffs are not distinguishable.