The Upsilon function from the filtered knot Floer complex

Let $K\subset S^3$ be a smooth oriented knot and $C={\mathrm{C}\mathrm{F}\mathrm{K}}^{\mathrm{\infty }}(K)$ a finitely generated free ${\mathbb{F}}_2[U,U^{-1}]$ complex. Choose a finite homogeneous basis $\{x_r\}$, with $M(x_r)$ and Alexander filtration $A(x_r)$. The translate $U^k{x}_r$ has Maslov grading $M(x_r)-2k$ and filtration coordinates $(i,j)=(-k,A(x_r)-k)$. The total homology is ${\mathbb{F}}_2[U,U^{-1}]$; use the standard normalization in which its Maslov-zero tower generator first appears at algebraic filtration level $i=0$, and denote it by ${\xi }_K$.

For $t\in [0,2]$ and a nonzero finite chain $z$, define $F_t(U^k{x}_r)=(1-t\mathrm{/}2)(-k)+(t\mathrm{/}2)(A(x_r)-k),F_t(z)=\max \{F_t(U^k{x}_r):U^k{x}_r\text{ has nonzero coefficient in }z\}\mathrm{.}$ Let ${\mathcal{Z}}_0(C)$ be the set of Maslov-zero cycles representing ${\xi }_K$. Set ${\nu }_t(C)={\min }_{z\in {\mathcal{Z}}_0(C)}{F}_t(z),{\mathrm{{\rm Y}}}_K(t)=-2{\nu }_t(C)\mathrm{.}$ The minimum is over cycles representing the normalized homology generator, not over individual basis terms. For comparison, let $C^{-}(K)$ be the ${\mathbb{F}}_2[U]$-span of the chosen basis and form its $t$-modified complex over the ring $\mathcal{R}$ of formal sums ${\sum }_{\alpha \in S}{c}_{\alpha }{v}^{\alpha }$ whose support $S\subset [0,\mathrm{\infty })$ is well-ordered (every nonempty subset of $S$ has a least element). The coefficient extension uses $U=v^2$. In the extended Laurent complex, $v$ lowers both filtration coordinates by $1\mathrm{/}2$ and Maslov grading by one, so a diagonal $U$-translation lowers the extended $t$-grading by two. On arbitrary homogeneous Laurent terms this grading is $M-t(j-i)$; the formula $M-tA$ below is specified on the chosen generators at $i=0$. A homogeneous generator $x$ has $t$-grading $M(x)-tA(x)$, extended by ${\mathrm{gr}}_t(v^{\alpha }x)={\mathrm{gr}}_t(x)-\alpha$. If a term $U^{k}y$ occurs in $\mathrm{\partial }x$, its $t$-modified differential coefficient is $v^{{\mathrm{gr}}_t(y)-{\mathrm{gr}}_t(x)+1}$; thus for $\mathrm{\partial }x={\sum }_y{c}_{xy}{U}^{(M(y)-M(x)+1)\mathrm{/}2}y$, define ${\mathrm{\partial }}_{t}x={\sum }_y{c}_{xy}{v}^{{\mathrm{gr}}_t(y)-{\mathrm{gr}}_t(x)+1}y$. This differential lowers $t$-grading by one. The comparison between its non-torsion tower grading and the finite cycle minimum is the separate filtered tower-splitting claim.

The real-exponent coefficient ring and its field of fractions

The diagram-specific knot Floer filtration and tau convention

Ozsvath, Stipsicz, and Szabo, Concordance homomorphisms from knot Floer homology, §§3–5

Livingston, Notes on the knot concordance invariant Upsilon, §§2–7

The diagramwise tau definition alone does not supply a normalized ${\mathrm{C}\mathrm{F}\mathrm{K}}^{\mathrm{\infty }}$ model. The current Grid homology is independent of the grid and computes knot Floer homology has an authored proof of the normalized grid-to-analytic comparison; localization gives the Laurent comparison under that supplier's stated prerequisites. The filtered tower-splitting item supplies the separate $t$-modified grading identification. This definition makes no assertion that $s=2\tau$ and does not itself prove model invariance.