Broken strings

Here we recall the definition of broken strings from [5], suitably modified for our purposes. Let K ⊂ R³ be a knot and p ∈ R³ be a point in the complement ofK. WriteQ=R³and viewQas the zero section inT^∗Q, and let N = LK ⊂T^∗Qbe the conormal bundle toK, whileLp is the cotangent fiber T_p^∗Q. We then have three Lagrangians Q, N, and Lp inT^∗Q; N and L_p are disjoint,QandL_p intersect transversely at p, and QandN intersect cleanly alongK. See Fig.4.

Fix base points(x0, ξ0)∈ N\K and(p, ξ) ∈ Lp\{p}. If we use a metric to identify T^∗QandT Q, then these points become(x0, v0)withv0 ∈ Tx₀N

Fig. 4 The cotangent bundleT^∗Qwith LagrangiansQ,N,L_p

and(p, v)withv ∈TpQ. This metric also gives a diffeomorphism between a neighborhood of the zero section in N (which in turn is diffeomorphic to all ofN) and a tubular neighborhood ofK inQ, and we can view Q∪N as the disjoint union ofQandN ⊂ Qglued alongK. This allows us to identifyTxN withTxQforx ∈ K. Similarly we view Q∪N ∪Lpas the disjoint union of Q∪N and Lp with p ∈ Q and 0∈ Lp identified, and the metric identifies T_pQwithT₀L_p =T_p^∗Q.

Now consider a piecewise C¹ path in Q ∪N ∪Lp. This path can move betweenQandN(in either direction) at a point onK = Q∩N, and between QandLpatp; we call the points where the path changes componentsswitches, either at K or at p.

Definition 3.1 Abroken stringis a piecewiseC¹paths : [a,b] → Q∪N∪Lp

such that:

• the endpointss(a),s(b)are each at one of the two base points(x0, v0)∈ N or(p, v)∈Lp;

• ifs(t₀)is a switch atKfromNtoQ(i.e., for small >0,s((t₀−,t₀])⊂N ands([t0,t0+))⊂Q), then:

lim

t→t₀⁻(s(t))^normal = lim

t→t₀⁺(s(t))^normal,

where we identifyT_s(t0)NwithT_s(t0)Qandv^normaldenotes the component ofvnormal to K with respect to the metric onQ;

• ifs(t0)is a switch at K fromQtoN, then:

lim

t→t0−(s(t))^normal = − lim

t→t0+(s(t))^normal;

• ifs(t0)is a switch at pfromLp toQ, then:

lim

t→t₀⁻s(t)= lim

t→t₀⁺s(t);

• ifs(t0)is a switch at pfrom QtoLp, then:

lim

t→t₀⁻s(t)= − lim

t→t₀⁺s(t).

The portions ofsinQ(respectivelyN,Lp) are calledQ-strings (respectively N-strings, L_p-strings).

Remark 3.2 A broken string models the boundary of a holomorphic disk in T^∗Qwith boundary onQ∪N∪L_p and one positive puncture at infinity at a Reeb chord forK∪p. The condition on the derivatives at a switch follows the behavior of the boundary of such a disk at a point where the boundary switches between QandN, or between QandLp: ifvinandvout denote the incoming and outgoing tangent vectors of a broken string at a switch then vout = Jvin, where J is the almost complex structure along the 0-section induced by the metric.

If we project fromT^∗QtoQ, then the endpoints of a broken string are each either at por at the point on K that is the projection ofx0. With this in mind, we call a broken strings:

• aK K broken string ifs(a)=s(b)=(x0, v0)

• aK pbroken string ifs(a)=(x0, v0)ands(b)=(p, v)

• a p K broken string ifs(a)=(p, v)ands(b)=(x0, v0)

• a ppbroken string ifs(a)=s(b)=(p, v). 3.2 String homology

We now construct a complex from broken strings whose homology might be called “string homology”; in Sect.3.4below, we will describe an isomorphism between this homology and enhanced knot contact homology.

For≥ 0, let denote the space of broken strings withswitches at p (note that we do not count switches atK here), equipped with theC^k-topology for somek ≥3. We write

=^{K K K pp K pp}

Fig. 5 The mapsδ_Q^K(respectivelyδ^K_N,δ_Q^p,δ_L^p

p) insert anN-string (Q-string,Lp-string, Q-string) at an interior point of aQ-string (N-string,Q-string,Lp-string) that lies onK (K, p,

p)

where ^{i j} denotes the subset of corresponding toi j broken strings for i, j ∈ {K,p}, and then

Ck()=C_k^{K K}()⊕C_k^{K p}()⊕C_k^{p K}()⊕C_k^pp()

for the freeZ-module generated by generick-dimensional singular simplices in(C_k^{i j}is the summand corresponding toi jbroken strings). Here “generic”

refers to simplices that satisfy the appropriate transversality conditions at switches and with respect to K and to p; compare [5, Definition 5.3].

In addition to the usual boundary operator∂ : Ck() → Ck−1() on singular simplices, there are two string operations

δ^KQ, δN^K : Ck()→Ck−1()

defined fork ≤2 in [5, Section 5.3] (where they are calledδQ,δN). We refer to [5] for details, but qualitatively these operations take a generick-dimensional family of broken strings, identify the subfamily consisting of broken strings where aQ-string orN-string has an interior point inK, and insert a “spike” in NorQat this point; see Fig.5. We note that this interior intersection condition is codimension 1, and that adding a spike increases the number of switches at K by 2. In our setting, there are two more string operations

δ_Q^p, δ_L^p_p : C_k()→C_k−2(+2)

that are defined in the same way asδ_Q^K, δ_N^K, but inserting spikes inL_p orQ where aQ-string orLp-string has an interior point atp; see Fig.5again. Note now that the interior intersection condition is codimension 2, and that adding a spike increases the number of switches at pby 2.

We then have the following result, which is a direct analogue of Proposi-tion 5.8 from [5] and is proved in the same way.

Lemma 3.3 On generic 2-chains, the operations∂,δ_Q^K +δ_N^K, andδ_Q^p +δ_L^p_p each have square 0 and pairwise anticommute. In particular, we have

∂+δ_Q^K +δ^K_N2

=0,

∂+δQ^K +δN^K δ_Q^p +δ_L^p_p +

δ_Q^p +δ_L^p_p ∂+δQ^K +δ^KN

=0, δ_Q^p +δ_L^p_p2

=0.

Lemma 3.3allows us to construct a complex out of broken strings in the following way. Form∈ ¹₂Z, define

C_m =

k+/2=m

C_k().

By consideration of the parity of the number of switches at p, we can write Cm =C_m^{K K} ⊕C_m^ppwhenmis an integer andCm =C_m^{K p}⊕C_m^{p K} whenmis a half-integer. We define a shifted complexC˜_∗,∗ ∈Z, by:

C˜_m^{K K} =C_m^{K K}, C˜_m^{K p}=C_m^{K p}_+1/2, C˜_m^{p K} =C_m^{p K}_−1/2, C˜_m^pp =C_m^pp, C˜m = ˜C_m^{K K} ⊕ ˜Cm^{K p}⊕ ˜Cm^{p K} ⊕ ˜Cm^pp;

that is, we shift the grading up by 1/2 if the beginning point is pand down by 1/2 if the endpoint is p. By Lemma3.3,∂+δ^K_Q +δ_N^K +δ_Q^p +δ_L^p_p is a differential onC˜_∗that lowers degree by 1.

Remark 3.4 The ¹₂-grading for strings broken atphas the following geomet-ric counterpart for holomorphic disks with switching Lagrangian boundary conditions onLp∪Qand a punctures at the intersection point p=Lp∩Q.

Consider a disku:(D, ∂D)→ (T^∗Q,L_p∪Q)withm punctures mapping to pand with a positive puncture asymptotic to a Reeb chorda. The formal dimension ofucan then be expressed as follows, see [4, Theorem A.1]:

dim(u)=(dim(Q)−3)+μ+m+1=μ+m+1, (2) whereμis the Maslov index of the loop of Lagrangian tangent planes along the boundary ofu. Here we close this loop by the capping operator ataand as follows at the punctures mapping to p: connect the incoming tangent plane (T_p^∗QorT^∗Lp) to the outgoing tangent plane (T^∗LporT_p^∗Q) with a negative

rotation along the Kähler angle (i.e. act bye^−iπ2s, 0 ≤ s ≤ 1). In the case at hand the tangent planes alongQ andLp are stationary with respect to the standard trivialization and the dimension formula reduces to

dim(u)= |a| +m −³₂+1