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Bigraded string types and their contributions to I^{bigr}

Statement

Assume AC (The Axiom of Choice) for the supplied well-definedness and isotopy invariance of ordinary and bigraded intersection numbers.

Fix bigradings ${\tilde{b}}_{k}, {\tilde{d}}_{k}$ of the basic arcs and vertical curves normalized by $I^{b i g r} ({\tilde{d}}_{k}, {\tilde{b}}_{k}) = 1 + q_{1}^{- 1} q_{2}, I^{b i g r} ({\tilde{b}}_{k}, {\tilde{b}}_{k + 1}) = 1 (0 \leq k \leq m in the first equation, 0 \leq k < m in the second),$ which determine the bigradings uniquely up to an overall shift $χ (r_{1}, r_{2})$ . For every admissible bigraded curve $\tilde{c}$ in normal form the bigraded intersection number $I^{b i g r} ({\tilde{b}}_{k}, \tilde{c})$ is computed by summing the contributions of the bigraded $k$ -strings of $\tilde{c}$ according to the following table: for $k > 0$ , $I_{0} (0, 0) \mapsto q_{1} + q_{2}, {I I}_{0} (0, 0) \mapsto q_{1} + q_{2}, {I I}_{0}^{'} (0, 0) \mapsto 1 + q_{1} q_{2}^{- 1}, {I I I}_{0} (0, 0) \mapsto q_{2}, {I I I}_{0}^{'} (0, 0) \mapsto 1,$ the exceptional types $I V, {I V}^{'}, V, V^{'}$ all contribute $0$ , and $V I (0, 0)$ contributes $1 + q_{2}$ ; a general type $I_{u} (r_{1}, r_{2})$ (and likewise for the other families) contributes the value of its $u = 0$ member multiplied by $q_{1}^{r_{1}} q_{2}^{r_{2}} (q_{1}^{- 1} q_{2})^{u}$ ; for $k = 0$ the zero-parameter types $V I I (0, 0), V I I I (0, 0), I X (0, 0), X (0, 0), X I (0, 0)$ contribute $0$ , $q_{1} q_{2}^{- 1} + 1$ , $1$ , $q_{1} q_{2}^{- 1} + 1$ , $1$ respectively; a type with parameters $(r_{1}, r_{2})$ has this value multiplied by $q_{1}^{r_{1}} q_{2}^{r_{2}}$ . In particular, $V I (r_{1}, r_{2})$ means $χ (r_{1}, r_{2}) {\tilde{b}}_{k}$ and $X I (r_{1}, r_{2})$ means $χ (r_{1}, r_{2}) {\tilde{b}}_{0}$ , as in the source’s printed p. 32.

Facts & Assumptions

Given: AC, the fixed standard picture, the normalized bigradings ${\tilde{b}}_{k}, {\tilde{d}}_{k}$ , a bigraded curve $\tilde{c}$ in normal form, and the type tables of Figures 15-18 together with the local index decorations of Figures 12-14.

[A1]

AC is inherited for the representative-independence and isotopy-invariance assertions used in [L2] and [L3] (The Axiom of Choice); the normalization of the fixed arcs and the finite local-index computations require no additional choice.

[L1]

In the standard picture $d_{k}$ crosses $b_{k}$ once in the interior and adjacent basic arcs share one marked endpoint (Basic arcs, admissible curves and the standard normal form). Bigradings of these arcs exist and differ by unique deck elements (Existence and rigidity of bigradings).

[L2]

Under AC, ordinary intersection weights add over the $k$ -strings in their minimal models (String types and their contributions to geometric intersection numbers). In such a model, each unmarked intersection contributes $(1 + q_{1}^{- 1} q_{2}) q_{1}^{μ_{1}} q_{2}^{μ_{2}}$ and each marked endpoint contributes $q_{1}^{μ_{1}} q_{2}^{μ_{2}}$ (Local indices and bigraded intersection numbers). The relative minimal-position construction fixing the other dividing arcs is the one in Khovanov–Seidel's proof of Lemma 3.18, printed pp. 29–31.

[L3]

For $k > 0$ , the preferred half-twist lift satisfies ${\tilde{t}}_{k} ({\tilde{b}}_{k}) = χ (- 1, 1) {\tilde{b}}_{k}$ , only asserting a deck shift for its supporting arc (The preferred lift of a half twist shifts the bigrading by chi(-1,1)). Under AC for the supplied bigraded intersection-number invariance, simultaneous transport preserves local indices, while shifting the second bigrading by $χ (r_{1}, r_{2})$ , or the first by $χ (- r_{1}, - r_{2})$ , multiplies each contribution by $q_{1}^{r_{1}} q_{2}^{r_{2}}$ (Local indices and bigraded intersection numbers).

[L4]

The $k$ -string $I_{u + 1} (r_{1}, r_{2})$ is obtained from $I_{u} (r_{1}, r_{2})$ by applying the half twist about $b_{k}$ ; the same holds for the other families, and the exceptional types are fixed or shifted according to Figure 15 (Basic arcs, admissible curves and the standard normal form).

Proof

technique · direct

1.1L1L2L3

Normalization. Each $d_{k}, b_{k}$ pair has one interior crossing, so its bigraded value is a monomial times $1 + q_{1}^{- 1} q_{2}$ ; each defined adjacent pair $b_{k}, b_{k + 1}$ has one common marked endpoint, so its value is a monomial. Choose the bigrading of $b_{0}$ , then successively shift $b_{k + 1}$ to make each adjacent monomial $1$ , and finally shift each $d_{k}$ to make its crossing monomial $1$ . Deck freeness makes these relative shifts unique. All simultaneous shifts of both families cancel in the relative indices and preserve these equations; hence the only ambiguity is a common overall shift. A shift of $\tilde{c}$ alone multiplies its contributions by the stated monomial. At $q_{1} = q_{2} = 1$ the normalizations are $2 = 2 I (d_{k}, b_{k})$ and $1 = 2 I (b_{k}, b_{k + 1})$ , consistent with the ordinary half weights.

1.2L2L4

The tables at the base parameters. Put each string into the relative minimal model used in the ordinary contribution lemma. The clockwise local-index paths in the decorated Figures 12–18 give, for $k > 0$ , an interior index $(1, 0)$ for $I_{0}, {I I}_{0}$ , an interior index $(1, - 1)$ for ${I I}_{0}^{'}$ , and a marked-end index $(0, 1)$ for ${I I I}_{0}$ and $(0, 0)$ for ${I I I}_{0}^{'}$ . The types $I V, {I V}^{'}, V, V^{'}$ are disjoint from $b_{k}$ . Multiplying interior monomials by $1 + q_{1}^{- 1} q_{2}$ gives $q_{1} + q_{2}, q_{1} + q_{2}, 1 + q_{1} q_{2}^{- 1}$ ; the single marked-end monomials give $q_{2}, 1$ . For $V I (0, 0) = {\tilde{b}}_{k}$ , a minimal self push-off has marked-end indices $(0, 0), (0, 1)$ , hence value $1 + q_{2}$ . For $k = 0$ , the positive boundary push gives no intersection for $V I I$ , an interior index $(1, - 1)$ for $V I I I, X$ , and a marked-end index $(0, 0)$ for $I X, X I$ , yielding $0, 1 + q_{1} q_{2}^{- 1}, 1, 1 + q_{1} q_{2}^{- 1}, 1$ . These are the local-index readings in the source’s bigraded string table; the ordinary relative minimal-position construction assigns each intersection to its string and realizes all model counts simultaneously.

2.1step 1.2L3L4

The family index and deck parameters. Write $C ({\tilde{b}}_{k}, \tilde{g})$ for the local contribution of a bigraded $k$ -string. Transporting its local-index paths by the preferred half twist and using [L3] gives $C ({\tilde{b}}_{k}, {\tilde{t}}_{k} \tilde{g}) = C ({\tilde{t}}_{k}^{- 1} {\tilde{b}}_{k}, \tilde{g}) = C (χ (1, - 1) {\tilde{b}}_{k}, \tilde{g}) = (q_{1}^{- 1} q_{2}) C ({\tilde{b}}_{k}, \tilde{g})$ . Normalization after twisting preserves these local contributions. By [L4], an integer-indexed type with index $u$ is the $u$ -th preferred half-twist iterate of its zero-index member; iteration gives $(q_{1}^{- 1} q_{2})^{u}$ , also for negative $u$ by inverting the monomial. The deck parameters add $q_{1}^{r_{1}} q_{2}^{r_{2}}$ by [L3]. Thus every indexed family has the claimed factor, including the shorter list for $k = m$ . Exceptional types and $k = 0$ types have only the deck factor. This uses the shift of the fixed $b_{k}$ , not a deck-shift assertion about $\tilde{g}$ .

3.1A1step 1.1step 1.2step 2.1∎

Conclusion. The bigraded intersection number $I^{b i g r} ({\tilde{b}}_{k}, \tilde{c})$ is the sum over the bigraded $k$ -strings of the contributions of the table, and the two normalizing equations determine the two families of bigradings up to an overall deck shift. AC is inherited for the supplied intersection-number invariance; the normalization and finite table calculations require no additional choice.

Depends on

Used by

The complex of an admissible bigraded curve Definition
Homs compute bigraded arc intersections Theorem

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mikhail Khovanov and Paul Seidel, Quivers, Floer Cohomology, and Braid Group Actions, J. Amer. Math. Soc. 15 (2002) 203-271, Lemma 3.20 (standard reference, not scraped)