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Raw tilde grid homology depends on the unknot grid
Statement refuted
Assume countable choice. The raw tilde grid homology is not a knot invariant by itself. The two specified grid diagrams both represent the oriented unknot, but their tilde homology ranks are and . For these grids the grid-to-knot-Floer comparison gives , where , , and by the standard genus-zero diagram and pointed-diagram invariance. Removing the factor would therefore falsely identify different raw ranks with the same rank-one knot invariant.
Statement references
The 2-by-2 finite rectangle calculation
The 3-by-3 finite rectangle calculation
The standard one-pointed unknot calculation
The grid-to-knot-Floer comparison
Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, Proposition 4.6.15
Manolescu, Ozsvath, and Sarkar, “A combinatorial description of knot Floer homology,” §4
Depends on
- Toroidal grid diagrams and gradings for oriented knots
- Finite grid complexes for knot Floer theory
- The three grid rectangle differentials square to zero
- The 2-by-2 unknot grid has tilde homology of rank two
- A 3-by-3 unknot grid has tilde homology of rank four
- Grid homology is independent of the grid and computes knot Floer homology
- The standard genus-zero hat diagram of the unknot has rank one
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- Grid moves generate isotopy of oriented knots
- Invariance of the multipointed Heegaard Floer complex
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links, Exercise 4.4.3 and Proposition 4.6.15 (standard reference, not scraped)
- Ciprian Manolescu, Peter Ozsvath, and Sucharit Sarkar, A combinatorial description of knot Floer homology, §4 (standard reference, not scraped)