Betti-Curve Representation

• Betti‐curve representation is a method that encodes families of Betti numbers as curves indexed by parameters, capturing homological features across various disciplines.
• It transforms complex homological data from algebraic, topological, or discrete settings into computationally tractable summaries for efficient analysis.
• Its applications span canonical tetragonal curves, persistent homology in TDA, configuration spaces, and discrete curvature theory, offering cross-disciplinary insights.

Searching arXiv for the cited papers and the term to ground the article in current arXiv records. arxiv_search(^^^^^^^^6query6 Curves^^^^^^^^6\6 OR ^^^^^^^^6\6 curve^^^^^^^^6\6 OR ^^^^^^^^6\6 representation6^^^^^^^^", max_results=^^^^6 OR \6\6^^^^) arxiv_search({"^^^^^^^^6query6 curve^^^^^^^^6\6 OR \6\6^^^^}) In the cited literature, the expression Betti-curve^^^^^^^^6 representation6^^^^^^^^ appears in several mathematically distinct settings. In algebraic geometry it denotes a combinatorial encoding of the graded Betti table of a canonical tetragonal curve by Newton-polygon data; in topology and topological data analysis it denotes Betti numbers organized as functions of a filtration, threshold, or discretized scale parameter; and in discrete curvature theory it can be viewed as a parameterized description of admissible first Betti numbers under curvature and degree constraints. A plausible unifying description is that a Betti-curve^^^^^^^^6 representation6^^^^^^^^ replaces a geometric, algebraic, or statistical object by Betti data indexed by an auxiliary parameter, or by simpler combinatorial data from which that Betti data is recovered.

^^^^6 OR \6^^^^. Terminological scope

The cited uses of the term are not identical, but they share a common formal pattern: a family of Betti numbers, or a Betti table, is represented through a one-parameter profile or through auxiliary combinatorial structure.

Setting	Parameter or encoding data	Betti object
Canonical tetragonal curves	Newton polygon and counts PRESERVED_PLACEHOLDER_^^^^6\6^^^^	Graded Betti table
Ordered configuration spaces	Number of points PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^	PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^
Symmetric matrices	Edge-density parameter PRESERVED_PLACEHOLDER_^^^^6 representation6^^^^	PRESERVED_PLACEHOLDER_^^^^6query6^^^^
Gaussian fields	Threshold PRESERVED_PLACEHOLDER_^^^^6\6^^^^	$\beta_k(\nu)$
Persistent diagrams	Filtration bins $\tau_i$	Betti sequence
Curved graphs	Degree and curvature constraints	Bounds on $\beta_1$

This distribution of meanings indicates that Betti-curve^^^^^^^^6 representation6^^^^^^^^ is best understood as a family of related constructions rather than a single standardized definition. In each case, the^^^^^^^^6 representation6^^^^^^^^ is designed to make homological information computable, comparable, or structurally transparent (&&&^^^^6\6^^^^&&&, &&&^^^^6 OR \6^^^^&&&, &&&^^^^6 OR \6^^^^&&&, &&&^^^^6 representation6^^^^&&&, &&&^^^^6query6^^^^&&&, &&&^^^^6\6^^^^&&&).

^^^^6 OR \6^^^^. Combinatorial syzygy encoding for canonical tetragonal curves

In the algebro-geometric setting, the notion arises from the study of canonical tetragonal curves. A smooth projective curve $C$ of genus PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^ is tetragonal if it carries a linear series PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^, and for a non-hyperelliptic curve the canonical embedding realizes PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^ as a canonical curve in PRESERVED_PLACEHOLDER_^^^^6 OR \6 representation6^^^^. Writing PRESERVED_PLACEHOLDER_^^^^6 OR \6query6^^^^ and PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^ for the canonical ideal, the minimal free resolution of PRESERVED_PLACEHOLDER_^^^^6 OR \66^^^^ defines graded Betti numbers PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^7, collected in the Betti table. Schreyer’s theorem states that for canonical tetragonal curves this entire graded Betti table is determined by two integers PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^8, Schreyer’s tetragonal invariants; conversely, the Betti table determines PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^9. Geometrically, after choosing a tetragonal pencil, the canonical curve lies on a rational normal threefold scroll PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^, and on the associated PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^ its strict transform is a complete intersection of two surfaces

PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^

with

PRESERVED_PLACEHOLDER_^^^^6 OR \6 representation6^^^^

For curves on toric surfaces, Castryck and Cools show that these invariants admit a purely combinatorial description in terms of the Newton polygon PRESERVED_PLACEHOLDER_^^^^6 OR \6query6^^^^ of a defining Laurent polynomial. If PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^ is the convex hull of the interior lattice points of PRESERVED_PLACEHOLDER_^^^^6 OR \66^^^^, PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^7, and

PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^8

then for a non-degenerate tetragonal curve PRESERVED_PLACEHOLDER_^^^^6 OR \6^^^^9,

PRESERVED_PLACEHOLDER_^^^^6 representation6\6^^^^

If moreover PRESERVED_PLACEHOLDER_^^^^6 representation6 OR \6^^^^, then Schreyer’s surface PRESERVED_PLACEHOLDER_^^^^6 representation6 OR \6^^^^ coincides with PRESERVED_PLACEHOLDER_^^^^6 representation6 representation6^^^^. This yields the pipeline

PRESERVED_PLACEHOLDER_^^^^6 representation6query6^^^^

The same paper pushes the idea further by addressing intrinsicness. If PRESERVED_PLACEHOLDER_^^^^6 representation6\6^^^^ and PRESERVED_PLACEHOLDER_^^^^6 representation66^^^^ are projectively equivalent in PRESERVED_PLACEHOLDER_^^^^6 representation6^^^^7, then PRESERVED_PLACEHOLDER_^^^^6 representation6^^^^8. In the tetragonal width-PRESERVED_PLACEHOLDER_^^^^6 representation6^^^^9 setting, this produces explicit conditions under which PRESERVED_PLACEHOLDER_^^^^6query6\6^^^^ is intrinsic to the curve. In those cases the interior polygon, the Schreyer invariants, and the canonical Betti table determine one another. This is the most literal instance in the cited literature of a Betti-curve^^^^^^^^6 representation6^^^^^^^^ as a combinatorial encoding of syzygies (&&&^^^^6\6^^^^&&&).

^^^^6 representation6^^^^. Persistent-homological^^^^^^^^6 representation6^^^^^^^^s of matrices and barcodes

In topological data analysis, a Betti-curve^^^^^^^^6 representation6^^^^^^^^ is a filtration-indexed family of ordinary Betti numbers. For a real symmetric PRESERVED_PLACEHOLDER_^^^^6query6 OR \6^^^^ matrix PRESERVED_PLACEHOLDER_^^^^6query6 OR \6^^^^, one first discards absolute values and retains only the relative order of the off-diagonal entries. The ordering matrix PRESERVED_PLACEHOLDER_^^^^6query6 representation6^^^^ determines a graph filtration PRESERVED_PLACEHOLDER_^^^^6query6query6^^^^, PRESERVED_PLACEHOLDER_^^^^6query6\6^^^^, by adding edges in increasing order of the corresponding matrix entries. Passing to the clique complex PRESERVED_PLACEHOLDER_^^^^6query66^^^^ gives a simplicial filtration, and the Betti curves are

PRESERVED_PLACEHOLDER_^^^^6query6^^^^7

These invariants depend only on the relative ordering of the off-diagonal entries and are invariant under simultaneous permutation of rows and columns. For rank-one symmetric matrices the constraints are especially rigid. If PRESERVED_PLACEHOLDER_^^^^6query6^^^^8 is positive rank one or negative rank one, then PRESERVED_PLACEHOLDER_^^^^6query6^^^^9 for all PRESERVED_PLACEHOLDER_^^^^6\6\6^^^^. In the positive mixed-sign rank-one case, if PRESERVED_PLACEHOLDER_^^^^6\6 OR \6^^^^ has PRESERVED_PLACEHOLDER_^^^^6\6 OR \6^^^^ negative entries, then

PRESERVED_PLACEHOLDER_^^^^6\6 representation6^^^^

with equality when PRESERVED_PLACEHOLDER_^^^^6\6query6^^^^ is the complete bipartite graph PRESERVED_PLACEHOLDER_^^^^6\6\6^^^^, and PRESERVED_PLACEHOLDER_^^^^6\66^^^^ for all PRESERVED_PLACEHOLDER_^^^^6\6^^^^7. In the negative mixed-sign case, again PRESERVED_PLACEHOLDER_^^^^6\6^^^^8 for all PRESERVED_PLACEHOLDER_^^^^6\6^^^^9 (&&&^^^^6 OR \6^^^^&&&).

A closely related, but distinct,^^^^^^^^6 representation6^^^^^^^^ is the Betti sequence derived from a persistence barcode or persistence diagram. After discretizing the filtration interval into bins, one defines a vector whose $\beta_k(\nu)$^^^^6\6^^^^-th entry counts the number of bars alive in the $\beta_k(\nu)$^^^^6 OR \6^^^^-th bin. The paper on stability shows that this naive Betti sequence is unstable with respect to the $\beta_k(\nu)$^^^^6 OR \6^^^^-Wasserstein metric: there exist persistence diagrams $\beta_k(\nu)$^^^^6 representation6^^^^ such that no non-negative constant $\beta_k(\nu)$^^^^6query6^^^^ satisfies

$\beta_k(\nu)$^^^^6\6^^^^

The instability comes from hard bin boundaries in birth-death space. To address this, the authors introduce a Gaussian-smoothed stabilized Betti sequence $\beta_k(\nu)$6 and prove a Lipschitz estimate

$\beta_k(\nu)$7

together with a normalized cumulative Betti sequence obtained by cumulative summation and $\beta_k(\nu)$8-normalization. This version turns barcodes into standard numeric feature vectors while preserving an explicit connection to persistent homology (&&&^^^^6query6^^^^&&&).

These two TDA uses share the same core idea: the homology of a filtration is summarized not by individual persistence intervals alone, but by the time-dependent ranks of homology groups.

^^^^6query6^^^^. Polynomial Betti curves for configuration spaces on an elliptic curve

For ordered configuration spaces, the relevant variable is not a filtration threshold but the number of particles. Let $\beta_k(\nu)$9 be a smooth complex elliptic curve and

$\tau_i$^^^^6\6^^^^

Fixing $\tau_i$^^^^6 OR \6^^^^, one studies

$\tau_i$^^^^6 OR \6^^^^

as a function of $\tau_i$^^^^6 representation6^^^^. In this setting, plotting $\tau_i$^^^^6query6^^^^ yields what the paper explicitly calls a Betti curve. The main asymptotic statement is that the $\tau_i$^^^^6\6^^^^-th Betti number grows as a polynomial of degree exactly $\tau_i$6. The algebraic engine is a differential bigraded algebra $\tau_i$7, the Križ model, together with an $\tau_i$8-module structure and a canonical filtration $\tau_i$9 by monomials using at most $\beta_1$^^^^6\6^^^^ distinct indices. When $\beta_1$^^^^6 OR \6^^^^, the differential is strict with respect to this filtration, and the associated graded dga $\beta_1$^^^^6 OR \6^^^^ computes the associated graded of the cohomology. Specializing to the elliptic curve $\beta_1$^^^^6 representation6^^^^, the paper derives polynomial expressions for mixed Hodge numbers and hence for Betti numbers (&&&^^^^6 OR \6^^^^&&&).

The paper also records explicit formulas for low cohomological degrees: $\beta_1$^^^^6query6^^^^ More generally, for $\beta_1$^^^^6\6^^^^,

$\beta_1$6

This establishes a Betti-curve^^^^^^^^6 representation6^^^^^^^^ in which homological complexity is encoded by explicit polynomials in the configuration size.

^^^^6\6^^^^. Threshold-dependent Betti curves of Gaussian random fields

In the study of Gaussian random fields, Betti curves are functions of a threshold level. For a smooth Gaussian random field $\beta_1$7, normalized by mean $\beta_1$8 and rms fluctuation $\beta_1$9, the excursion set at threshold $C$^^^^6\6^^^^ is

$C$^^^^6 OR \6^^^^

In dimension three, the nontrivial Betti numbers of $C$^^^^6 OR \6^^^^ are $C$^^^^6 representation6^^^^, interpreted respectively as the number of connected excursion regions, the number of circular holes or tunnels, and the number of three-dimensional voids. In dimension two, the relevant Betti numbers are $C$^^^^6query6^^^^ and $C$^^^^6\6^^^^. The Betti-curve^^^^^^^^6 representation6^^^^^^^^ is precisely the family $C$6 as $C$7 varies (&&&^^^^6 representation6^^^^&&&).

A central point is the relation to the Euler characteristic and genus. In three dimensions,

$C$8

while in two dimensions

$C$9

Thus the genus curve is a signed sum of Betti curves. The paper emphasizes that each Betti number dominates the genus in a different threshold regime: PRESERVED_PLACEHOLDER_^^^^6 OR \6\6\6^^^^ dominates the high-threshold region and measures cluster abundance, PRESERVED_PLACEHOLDER_^^^^6 OR \6\6 OR \6^^^^ dominates near median thresholds and measures tunnel-rich or sponge-like topology, and PRESERVED_PLACEHOLDER_^^^^6 OR \6\6 OR \6^^^^ dominates the low-threshold region and measures void abundance. Unlike the Gaussian genus curve, whose shape is fixed for all Gaussian fields regardless of power spectrum, both the amplitude and shape of the Gaussian Betti curves depend on the slope PRESERVED_PLACEHOLDER_^^^^6 OR \6\6 representation6^^^^ of the power spectrum. The curves become broader and their amplitudes drop less steeply than the genus as PRESERVED_PLACEHOLDER_^^^^6 OR \6\6query6^^^^ decreases.

This use of Betti-curve^^^^^^^^6 representation6^^^^^^^^ is therefore a threshold-resolved topological summary of a random field. The paper presents it as a more elaborate characterization of topology than genus alone, while also noting the practical inconvenience that Gaussian Betti curves must be computed separately for each power spectrum.

6. Curvature-constrained Betti profiles on graphs

In discrete curvature theory, the phrase is less literal, but the cited exposition treats first Betti number bounds as a kind of parameterized Betti profile. For a finite graph with non-negative Ollivier curvature, the first Betti number of the PRESERVED_PLACEHOLDER_^^^^6 OR \6\6\6^^^^-complex PRESERVED_PLACEHOLDER_^^^^6 OR \6\66^^^^ satisfies

PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^7

If there exists a vertex PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^8 such that

PRESERVED_PLACEHOLDER_^^^^6 OR \6\6^^^^9

then

PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6\6^^^^

The extremal case is rigid: non-negative Ollivier curvature together with

PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 OR \6^^^^

is equivalent to PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 OR \6^^^^ being a discrete flat torus. The same Ollivier-curvature results extend to potentially non-reversible Markov chains with symmetric support, and the paper also treats bone-idle graphs, where PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 representation6^^^^ for all PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6query6^^^^ and all edges (&&&^^^^6\6^^^^&&&).

For non-negative Bakry-Émery curvature, the paper studies the first Betti number of the path-homology PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6\6^^^^-complex PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \66^^^^ and proves

PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^7

It also establishes a defect bound when non-negative Ollivier curvature is required only outside a finite subset PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^8: PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6^^^^9 For cycles of length at least five, equipped with the unique path metric with constant Ollivier curvature, the upper Betti number bound is attained if and only if the curvature is zero.

This suggests a broadened meaning of Betti-curve^^^^^^^^6 representation6^^^^^^^^: not only a filtration-indexed function, but also an upper-envelope description of allowable Betti numbers under structural constraints such as curvature, degree, and idleness. In that sense, the paper gives a discrete Bochner-type parameterization of homological complexity.

7. Conceptual synthesis

Across these settings, Betti-curve^^^^^^^^6 representation6^^^^^^^^ serves three recurrent purposes. First, it acts as a computational reduction: Newton polygons determine Schreyer invariants and hence Betti tables; barcodes are converted into fixed-length vectors; and Gaussian fields are reduced to threshold-indexed topological summaries. Second, it acts as a structural invariant: the^^^^^^^^6 representation6^^^^^^^^ depends on order type for symmetric matrices, on interior-polygon combinatorics for tetragonal toric curves, on the filtration variable for configuration spaces, and on curvature constraints for graphs. Third, it acts as a comparative device: one compares shapes of PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6\6^^^^, PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 OR \6^^^^, or polynomial growth laws PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 OR \6^^^^ rather than raw geometric objects.

The cited literature therefore does not present a single theory of Betti-curve^^^^^^^^6 representation6^^^^^^^^. Instead, it presents a family of mathematically parallel constructions whose common content is the organization of homological information into a curve-like, parameterized, or combinatorially recoverable form. In algebraic geometry the output is a Betti table; in TDA it is a filtration-indexed sequence or stabilized vectorization; in random-field topology it is a threshold-dependent set of Betti curves; in configuration-space cohomology it is a polynomial function of PRESERVED_PLACEHOLDER_^^^^6 OR \6 OR \6 representation6^^^^; and in graph curvature it is a degree-curvature envelope for the first Betti number. This suggests that the term is best regarded as a cross-disciplinary template for encoding topology through Betti data, rather than as a uniquely fixed technical term.