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Homs compute bigraded arc intersections

Statement

Assume AC, inherited from the representative-independence and isotopy invariance of intersection numbers (Local indices and bigraded intersection numbers) and used to interpret $σ \in B_{m + 1}$ as a boundary-fixed mapping class acting on bigraded curves (Basic arcs, admissible curves and the standard normal form, with its Artin completeness and smooth comparison); the graded Hom and Poincaré-polynomial computation is finite. For all $σ, τ \in B_{m + 1}$ , all $s_{1}, s_{2} \in Z$ and all $0 \leq k, j \leq m$ the abelian group ${Hom}_{C_{m}} (R_{τ} P_{k}, R_{σ} P_{j} [s_{1}] {- s_{2}})$ is free, and its Poincaré polynomial satisfies $\sum_{s_{1}, s_{2}} rk {Hom}_{C_{m}} (R_{τ} P_{k}, R_{σ} P_{j} [s_{1}] {- s_{2}}) q_{1}^{s_{1}} q_{2}^{s_{2}} = I^{b i g r} ({\tilde{f}}_{τ} ({\tilde{b}}_{k}), {\tilde{f}}_{σ} ({\tilde{b}}_{j})),$ where ${\tilde{f}}_{σ}$ is the preferred lift of the boundary-fixed mapping class representing $σ$ under the isomorphism $B_{m + 1} ≅ G$ , acting on the normalized bigradings of the basic arcs. In particular, specializing $q_{1} = q_{2} = 1$ gives $I^{b i g r} (\dots) ∣_{q_{1} = q_{2} = 1} = 2 I (\cdot, \cdot),$ so the ranks at $q_{1} = q_{2} = 1$ recover twice the ordinary geometric intersection numbers of the source's curves.

Facts & Assumptions

Given: AC, the braid group action on bigraded curves by preferred lifts, the complexes $R_{σ}$ acting on $C_{m}$ , the normalized bigradings of the basic arcs, and the complex $L (\tilde{c})$ of an admissible bigraded curve.

[L1]

$R_{σ} P_{j} ≅ L (σ {\tilde{b}}_{j})$ for normalized bigradings of the basic arcs, and $R_{σ}$ is an equivalence of $C_{m}$ with inverse $R_{σ^{- 1}}$ (Curve complexes intertwine the braid generators, The Khovanov-Seidel complexes give a weak derived braid action).

[L2]

$L (\tilde{c})$ is a bounded complex of finite graded projectives and its homotopy type depends only on the isotopy class of $\tilde{c}$ ; the deck action acts by shifts $L (χ (r_{1}, r_{2}) \tilde{c}) ≅ L (\tilde{c}) [- r_{1}] {r_{2}}$ (The curve complex is a complex and is invariant under normal-form moves, The complex of an admissible bigraded curve).

[L3]

For chain complexes over an abelian category the homotopy classes of chain maps are the degree-zero homology of the Hom complex: ${Hom}_{K} (C, D) ≅ H_{0} (\underset{}{Hom} (C, D))$ , and the bigraded Hom groups of $C_{m}$ are the degree-zero homology of the bigraded Hom complex (Hom in the homotopy category is zero-degree homology of the Hom complex).

[L4]

Under AC, $I^{b i g r}$ is invariant under the preferred lifts: $I^{b i g r} (\tilde{f} ({\tilde{c}}_{0}), \tilde{f} ({\tilde{c}}_{1})) = I^{b i g r} ({\tilde{c}}_{0}, {\tilde{c}}_{1})$ , and it is a sum of the contributions of the $k$ -strings, with the table of Bigraded string types and their contributions to I^{bigr} (Local indices and bigraded intersection numbers).

[L5]

The graded maps $P_{k} = A_{m} e_{k} \to P_{j} = A_{m} e_{j}$ are right multiplication by paths in $e_{k} A_{m} e_{j}$ . The finite path basis gives zero when $∣ k - j ∣ > 1$ , one arrow for adjacent vertices, the vertex and degree-one return for $k = j > 0$ , and only the vertex for $k = j = 0$ . Thus adjacent and internally shifted self Homs must not be discarded (Finite graded A_m-modules, internal shifts and the vertex projectives, The 4m+1 path basis).

[L6]

Bigradings of non-closed curves exist and are unique up to the deck action, and ${\tilde{b}}_{j}$ is a basic arc (Existence and rigidity of bigradings, Basic arcs, admissible curves and the standard normal form).

[F1]

Literature input. KS Lemma 4.10 proves the string decomposition of shifted Homs: the only surviving projectives have $∣ x_{0} - k ∣ \leq 1$ , and the differential components joining different $k$ -strings induce zero on $Hom (P_{k}, -)$ . Lemmas 4.11–4.12 prove that each string Hom group is free, with the base Poincaré tables equal to the complete bigraded contribution tables, and the parameters multiply them by $q_{1}^{r_{1}} q_{2}^{r_{2}} (q_{1}^{- 1} q_{2})^{u}$ for the integer-indexed families. Their proof first removes deck shifts and integer twists and only then computes the base diagrams. For type VI(0,0), one has $L ({\tilde{b}}_{k}) = P_{k}$ ; its self-Hom has the vertex in bidegree $(0, 0)$ and the return in bidegree $(0, 1)$ , both with homological shift zero. Thus its polynomial is $1 + q_{2}$ , agreeing directly with the source tables and the self-intersection computation of [L4]. This is a stated literature input; we do not infer free homology merely from free chains (KS full proof, printed pp. 45–47).

Proof

technique · source-supported direct reduction

1.1L1L4L6

Reduction to the first untwisted arc. Apply the inverse equivalence $R_{τ^{- 1}}$ to both arguments of each shifted Hom. The weak action identifies the second image with $R_{τ^{- 1} σ} P_{j}$ . Apply the corresponding inverse preferred lift to both geometric arcs; [L4] gives the same reduction of their intersection polynomial. It therefore suffices to compare $Hom (P_{k}, L (\tilde{c}) [s_{1}] {- s_{2}})$ and $I^{b i g r} ({\tilde{b}}_{k}, \tilde{c})$ , with $\tilde{c} = {\tilde{f}}_{τ^{- 1} σ} {\tilde{b}}_{j}$ by [L1], a bigraded basic-arc image.

1.2L2L3L5F1

The correct string decomposition. Compute these groups as homology of the Hom complex by [L3]. The corner basis [L5] kills summands farther than one vertex from $k$ , while retaining adjacent arrows and the self return. For different $k$ -strings the remaining connecting differential acts by a length-three product or a forbidden return at the exterior vertex, hence is zero, as the source string-splitting calculation in [F1] proves. Thus for every pair of shifts the Hom group is the finite direct sum of the string Hom groups.

1.3L2L3L4F1

Base diagrams, integer twists and freeness. The source local theorem [F1] computes the BASE diagrams, then extends by deck shifts and integer half twists. For example its $I_{0}$ Hom complex reduces to $0 \to Z {1} \overset{0}{\to} Z \to 0$ , giving $q_{1} + q_{2}$ and free groups. The four exceptional zero-contribution types have acyclic Hom complexes; the other base types give the full table of [L4], with type VI checked directly in [F1]. The source's parameter reduction multiplies the table by the stated monomial, without claiming an arbitrary winding string has at most three terms. Crucially [F1] asserts freeness of the actual string HOMOLOGY groups; that property is not inferred from the chain groups alone.

2.1step 1.1step 1.2step 1.3L4F1

The polynomial and all shifts. Summing the string polynomials of step 1.3 gives $I^{b i g r} ({\tilde{b}}_{k}, \tilde{c})$ by [L4], while step 1.2 identifies each coefficient with the rank of the corresponding shifted Hom group. A finite direct sum of the free string Hom groups is free. Step 1.1 now returns the original $σ, τ$ and both geometric images, proving the claimed full Poincaré formula and freeness for every shift.

3.1step 2.1L4∎

Specialization and conclusion. Property (B1) of [L4] evaluates the polynomial at $q_{1} = q_{2} = 1$ as twice the ordinary intersection number. The equivalence reduction, correctly retained corner Homs, exact source string theorem and coefficient-wise freeness prove all claimed conclusions. AC is inherited through the braid/mapping-class dictionary and the supplied representative-independence and isotopy invariance of intersection numbers; the finite string groups add no choice requirement.

Depends on

Used by

The Khovanov-Seidel weak braid action is faithful Theorem

Dependency tree · two levels

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Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Section 4c, Proposition 4.9 and Theorem 1.1 (standard reference, not scraped)