modules to recover the Alexander moduleH1(X˜K)ofK, whereX˜Kis the infi-nite cyclic cover ofR3\KandH1(X˜K)is viewed as aZ[m±1]-module as usual by deck transformations. As a consequence, we show that a certain canoni-cal linearization of enhanced knot contact homology contains the Alexander module and thus the Alexander polynomial.
It was previously known [24] that the Alexander module can be extracted from the same linearization of usual knot contact homology LCH∗(K), but in a somewhat obscure way—essentially, the degree 1 linearized homology is the second tensor product of H1(X˜K)⊕Z[m±1], with the proof involving an examination of the combinatorial form of the DGA ofK in terms of a braid representative forK, and a relation to the Burau representation. Here we will see that with the introduction of the fiberp alongsideK, we can instead deduce the Alexander module in a significantly simpler way. In particular, we will use linearized homology not in degree 1 but in degree 0, which is more geometrically natural (for instance, it relates more easily to the cord algebra).
We first present a variant of the cord algebra and modules, following [24]
and especially the discussion in [5, §2.2]. Choose a base point∗on K corre-sponding to the base point∗onK. Let anunframed cordofKbe a path whose endpoints are in(K\{∗})∪ {p}and which is disjoint from K in its interior;
we can divide these into K K, K p, p K, pp cords depending on where the endpoints lie.
Definition 4.5 ([24]) Theunframed cord algebraofK, CordK K, is the non-commutative algebra over Z[l±1,m±1] generated by homotopy classes of unframedK K cords, modulo the following skein relations:
(1) = 1−m
(2) * =l·
* and
* =
* ·l
(3) −m = · .
The unframed Kp (respectively pK) cord module of K, CordK p (respec-tively Cordp K), is the right (respectively left) CordK K-module generated by unframedK p(respectively p K) cords, modulo the skein relations (2) and (3).
Note that CordK K, CordK p, and Cordp K are all Z[l±1,m±1]-modules, unlike their framed counterparts CordK K, CordK p, Cordp K, where elements ofZ[l±1,m±1]do not necessarily commute with cords. However, we have the following.
Proposition 4.6 The unframed cord algebra and modulesCordK K,CordK p, Cordp K are isomorphic to the quotients of the cord algebra and modules CordK K,CordK p,Cordp K obtained by imposing the relations that elements ofZ[l±1,m±1]commute with cords.
Proof This is essentially laid out in [5, §2.2]. Fix a cordγ0 from pto a point x0 ∈ K\{∗}. Given any cord γ, we can produce a loop γ˜ in R3\K based at p, by joining any endpoint ofγ on K tox0 along (any path in) K, and appendingγ0or−γ0as necessary. Letγbe the unframed cord obtained from γ by joining any endpoint ofγ on Kto the corresponding point on K by a straight line segment normal to K. Then the map
γ →m−lk(γ ,˜ K)γ
gives the desired isomorphisms from the quotients of CordK K, CordK p, Cordp K to CordK K, CordK p, Cordp K. (For the inverse maps from Cord to Cord, homotope any cord with a beginning or end point onK so that it begins or ends withγ0 or−γ0, and then remove±γ0.) Note that the displayed map from Cord to Cord sends the skein relations (1), (3), (4) in Definition4.2to (1), (2), (3) in Definition 4.5, and the normalization by powers of m means that (2) from Definition4.2becomes trivial under this map.
Now from [24], there is a canonical augmentation of the DGA forK, : (AK, ∂)→(Z[m±1],0),
whose definition we recall here. SinceAKis supported in nonnegative degree, the graded map is determined by its action on the degree 0 part ofAK, or equivalently (since◦∂ =0) by the induced action onH0(AK, ∂). This in turn is determined by the induced action on CordK K, which by Proposition4.6 is the quotient ofH0(AK, ∂)by settingl,mto commute with everything. On CordK K,is defined as follows:
(l)=1 (m)=m
(γ )=1−m
for any unframed K K cord γ. (Note that preserves the skein relations for CordK K and is thus well-defined.) We can extendfrom an augmentation of AK to an augmentation ofAK∪p by settingto be 0 for any mixed chord betweenK andp.
Remark 4.7 Applying [1, Theorem 6.15] to the holomorphic strips over binor-mal chords shows that the augmentationis induced by an exact Lagrangian filling MK diffeomorphic to the knot complement, obtained by joining the conormalLK and the zero-sectionQvia Lagrange surgery along the knotK. Linearizing with respect to this augmentation gives the linearized contact homology
LCH∗ K ∪p
= LCH∗
K,K ⊕ LCH∗
K,p ⊕ LCH∗
p,K ⊕ LCH∗
p,p. As discussed previously, in [24] it is shown that(LCH1)K,Krecovers the Alexander moduleH1(X˜K). Here instead we have the following.
Proposition 4.8 We have isomorphisms ofZ[m±1]-modules LCH0
K,p ∼= LCH1
p,K
∼= H1(X˜K)⊕Z[m±1].
Proof We will prove the isomorphism for (LCH1)p,K; the isomorphism for (LCH0)K,p follows by symmetry between CordK p and Cordp K. The complex whose homology computes(LCH∗)p,Kis the freeZ[m±1]-module generated by Reeb chords topfromK, with the differential∂lin given by
Fig. 7 Three p Kcordsγ1, γ2, γ3related byγ1−mγ2=(1−m)γ3
applying the augmentationto all pure Reeb chords fromK to itself to the usual differential∂. In particular, since the degree 1 homology(LCH1)p,K
is the quotient of theZ[m±1]-module generated by degree 1 Reeb chords to pfromK by the image of∂lin, we have:
(LCH1)p,K ∼= Rp K⊗ Z[m±1] ∼=Cordp K⊗Z[m±1].
Here by “⊗” we mean⊗RK K (or⊗CordK K) where we use to giveZ[m±1] the structure of an RK K-module (or CordK K-module), and implicitly we are settingl,mto commute with everything in(LCH1)p,K and Rp K.
Now Cordp K⊗Z[m±1] is the quotient of the free Z[m±1]-module gen-erated by unframed p K cords by the skein relations (2) and (3) from Definition4.5, wherelis sent to 1 and allK Kcords are sent to 1−m. Relation (2) then says that p K cords are unchanged if we move theirK endpoint over
∗, while relation (3) becomes:
−m = (1−m) .
That is, ifγ1, γ2, γ3 are unframed p K cords that are related as shown in the left side of Fig.7, then we impose the relation:
γ1−mγ2=(1−m)γ3.
Thus we can describe Cordp K⊗Z[m±1] in terms of a knot diagram for K as follows. Use the diagram to placeK in a neighborhood of thex yplane inR3, and place p high above the x y plane along thez axis. If the diagram hasncrossings, then it dividesK intonstrands from undercrossing to under-crossing. Then Cordp K⊗Z[m±1] is generated by n unframed p K cords, namely straight line segments from p to any point on each of these strands, and each crossing gives a relation γ1 −mγ2 = (1−m)γ3 ifγ1, γ2, γ3 are
as shown in the right side of Fig.7. But this is the well-known presentation of H1(X˜K)⊕Z[m±1]from knot colorings. In particular, what we have just described is the Alexander quandle ofK, see [19].
Remark 4.9 The description we have given in this section for the unframed cord modules is highly reminiscent of the construction of the knot quandle from [19], which is known to be a complete invariant. However, we do not know how to extract the entire knot quandle, rather than just the Alexander quandle (which is a quotient), from the unframed cord module.