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The three grid rectangle differentials square to zero
Statement
For every one-component toroidal grid and each of the tilde, associated-graded minus, and full filtered minus modes, the differential defined by the grid-complex algorithm satisfies over its stated coefficient ring.
Statement references
Facts & Assumptions
Given: A one-component toroidal grid and one of the three differential modes defined by the algorithm.
Finite grid complexes for knot Floer theory defines each differential as a finite sum over empty-state rectangles, with the stated mode-specific marking restriction and coefficient.
Rectangle-boundary pairings and grading changes in grid complexes proves that every non-annular two-rectangle composite has two equal contributions and that the only possible annular contributions either vanish by mode restriction or occur twice with the same monomial.
Proof
Fix initial and terminal states . The coefficient of in is the sum over all ordered pairs of counted empty rectangles with an intermediate state. By [F2], every non-annular composite contributes in a pair of decompositions with equal coefficient, and so its total is zero in .
If , all composites are non-annular and step 1.1 gives zero coefficient. If , the only additional composites are thin annuli; [F2] shows these are excluded in the first two modes and cancel in the full filtered mode. Thus every matrix coefficient of is zero, for every mode.
Depends on
Used by
- Raw tilde grid homology depends on the unknot grid Counterexample
- Worked grid-complex cases for the unknot Example
- A 3-by-3 unknot grid has tilde homology of rank four Lemma
- Basepoint variables act chain homotopically on a knot grid Lemma
- The 2-by-2 unknot grid has tilde homology of rank two Lemma
- Commutation and stabilization invariance of grid complexes Theorem
- Grid homology is independent of the grid and computes knot Floer homology Theorem
Dependency tree · two levels
4 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
- Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links, Lemma 4.6.7 and §13.2 (standard reference, not scraped)
- Ciprian Manolescu, Peter Ozsvath, and Sucharit Sarkar, A combinatorial description of knot Floer homology, §1 (standard reference, not scraped)