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Grid Upsilon from the diagonal grid complex

Definition

Let $G$ be an $n \times n$ toroidal grid diagram representing a knot. Let $G C^{-} (G)$ be the finite grid complex over $F_{2} [U_{1}, \dots, U_{n}]$ with its Maslov grading $M$ and Alexander filtration $A$ . Form the diagonal quotient $D_{G} = G C^{-} (G) / (U_{1} - U_{2}, \dots, U_{1} - U_{n}),$ regarded as a finitely generated $F_{2} [U]$ complex with $U = U_{1} = \dots = U_{n}$ .

For a finitely generated Maslov-graded, Alexander-filtered complex $D$ over $F_{2} [U]$ with a fixed finite homogeneous basis, its $t$ -modified complex for $t \in [0, 2]$ is the free $R$ -module on that basis, where $R$ is the ring of formal $F_{2}$ -series $\sum_{α \in A} c_{α} v^{α}$ with support $A \subset [0, \infty)$ such that every nonempty subset of $A$ has a least element. A homogeneous generator $x$ has $t$ -grading ${gr}_{t} (x) = M (x) - t A (x)$ , extended by ${gr}_{t} (v^{α} x) = {gr}_{t} (x) - α$ . The coefficient extension uses $U = v^{2}$ ; this action lowers the extended $t$ -grading by two. The formula $M - t A$ is specified on the chosen polynomial generators. On Laurent translates, the extended grading is $M - t (j - i)$ , with $v$ lowering both filtration coordinates by $1 / 2$ . If $\partial x = \sum_{y} c_{x y} U^{(M (y) - M (x) + 1) / 2} y,$ then the modified differential is $\partial_{t} x = \sum_{y} c_{x y} v^{{gr}_{t} (y) - {gr}_{t} (x) + 1} y .$ A homogeneous homology class is non-torsion if no nonzero element of $R$ annihilates it. For $0 \leq t \leq 1$ , define $Υ_{G} (t)$ to be the maximum $t$ -grading of a homogeneous non-torsion class in $H_{*} (D_{G}^{t})$ . For $1 < t \leq 2$ , define $Υ_{G} (t) = Υ_{G} (2 - t)$ . This is the grid convention compared with the original cycle-minimax invariant.

Definition references

The coefficient valuation ring

Finite homogeneous matrix reduction over that ring

Toroidal grid diagrams and their knot orientation convention

The finite grid chain complex and its marking variables

The original Upsilon cycle-minimax convention and Maslov/Alexander grading convention

Kubota, On the Upsilon invariant in grid homology, §§2.5 and 3

Ozsvath, Stipsicz, and Szabo, Grid Homology for Knots and Links, §13.2 and Lemma 14.1.11

Facts & Assumptions

Given: A finite grid diagram $G$ for a knot, its finite grid complex, and the coefficient and grading conventions specified in the definition.

[F1]

Toroidal grid diagrams and gradings for oriented knots: a knot grid has one $O$ and one $X$ in each row and column, with the induced oriented component permutation a single cycle.

[F2]

Finite grid complexes for knot Floer theory: the grid construction returns a finite state complex with the displayed marking-variable differential; its identification with knot Floer homology is a separate theorem.

[F3]

The Upsilon function from the filtered knot Floer complex: $t$ -grading is $M - t A$ , multiplication by $v^{α}$ lowers it by $α$ , and the modified differential is reweighted to have degree $- 1$ .

[F4]

Grid homology is independent of the grid and computes knot Floer homology now has an authored Proof section. Its Statement supplies the chosen-variable filtered comparison and the simultaneous-collapse $C F K^{-} (K) \otimes V^{\otimes (n - 1)}$ factor needed here. MOS Proposition 2.3 states the chosen-variable filtered comparison and the tilde-homology $V$ -factor, and Theorem 3.3 identifies the grid filtered complex with the knot filtration; the current in-run proof supplies the chain-level ordinary-collapse factor and the absolute normalization under its stated prerequisites. Under that comparison, the normalized $C F K^{-}$ tower gives a non-torsion summand in the diagonal quotient's $t$ -modified homology.

Proof

1.1givenF2F3algebra

Every term in $\partial_{t}^{2} x$ corresponding to a two-step path $x \to y \to z$ has exponent $({gr}_{t} (y) - {gr}_{t} (x) + 1) + ({gr}_{t} (z) - {gr}_{t} (y) + 1) = {gr}_{t} (z) - {gr}_{t} (x) + 2$ , which is independent of the intermediate generator $y$ . Its coefficient is therefore the coefficient of $z$ in $\partial^{2} x$ , multiplied by that common power of $v$ . Since $\partial^{2} = 0$ , $\partial_{t}^{2} = 0$ . For every differential term the exponent is nonnegative: writing its $U$ -power as $k$ , the Maslov relation gives $M (y) - M (x) + 1 = 2 k$ , and the Alexander filtration gives $A (y) - A (x) \leq k$ , hence $2 k - t (A (y) - A (x)) \geq 0$ for $0 \leq t \leq 2$ . Thus $D_{G}^{t}$ is a chain complex over $R$ .

2.1givenF1F2F3F4step 1.1algebra

The quotient identifies finitely many marking variables in a complex with finitely many grid states, so $D_{G}$ has a finite polynomial basis. Its t-modification is a finite homogeneous complex over the coefficient ring proved in The real-exponent coefficient series form a valuation domain. The explicit finite matrix reduction of Finite homogeneous complexes over the real-series ring have finite cyclic splittings gives a finite sum of free homology generators and cyclic torsion summands, and proves that, if a free summand is present, its largest homogeneous non-torsion grading exists. Thus the asserted maximum is justified without a Noetherian assumption on $R$ . The current grid comparison [F4] supplies a nonzero free tower for this actual diagonal quotient.

3.1givenF1F2step 2.1∎

The quotient convention and the symmetric extension for $t > 1$ are part of the definition, not consequences of the comparison theorem. No grid-independence or agreement with the knot Floer Upsilon invariant is asserted here; those are theorems on this page.

Unresolved proof obligations

The current grid-homology comparison theorem now has an authored proof and supplies the non-torsion tower needed to make this maximum well-defined, under its stated analytic and normalization prerequisites. This definition itself still makes no grid-independence or Upsilon-agreement claim; those are supplied by the theorem on the assigned page.

Depends on

Used by

The grid Upsilon function agrees with the original Upsilon invariant Theorem

Dependency tree · two levels

21 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

Hajime Kubota, On the Upsilon invariant in grid homology, §§2.5 and 3 (standard reference, not scraped)
Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo, Grid Homology for Knots and Links, §13.2 and Lemma 14.1.11 (standard reference, not scraped)