Betti Number Classification

• Betti number classification is a framework that uses numerical invariants from algebraic and topological settings to organize and distinguish complex structures.
• It employs methods from graded free resolutions, combinatorial graph reductions, and asymptotic quasi-polynomial analysis to reveal forbidden regions and explicit classification formulas.
• The approach has practical applications across commutative algebra, geometry, topological data analysis, and machine-learning, offering both exact and approximate classifications.

Searching arXiv for papers on Betti-number classification and related topics. Betti number classification is not a single theorem but a family of classification programs in which Betti numbers, or their graded and multigraded refinements, are used to organize algebraic, combinatorial, geometric, and applied objects. In commutative algebra, the problem concerns the shape and occurrence of Betti tables in minimal free resolutions; in graph theory, it may mean classification by the first Betti number $b_1$, or cycle rank; in geometry, it includes rigidity statements in which a Betti number or an extremal Betti entry determines a homeomorphism type or a syzygetic class; and in topological data analysis it appears as a discrete but incomplete invariant for multiparameter persistence modules (Pour, 2015, Kramer, 1 Jul 2025, Choi et al., 2010, Moore, 2020).

1. Multiple meanings of “Betti number”

The term “Betti number” carries different technical meanings across the literature. In graph theory, the first Betti number is the cyclomatic number

$b_1=|E|-|V|+b_0,$

and for connected cubic graphs this becomes $b_1=\frac12|V|+1$ (Kramer, 1 Jul 2025). In graded commutative algebra, the basic objects are graded Betti numbers

$$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$

which record the number of generators in degree $j$ at homological degree $i$ in a minimal graded free resolution (Pour, 2015). In the multigraded setting one instead studies $\beta_{i,\alpha}(L)$, indexed by multidegrees $\alpha\in \mathbb Z^m$ (Charalambous et al., 2010). In the Hodge-theoretic setting of compact Riemannian manifolds, the classical Betti numbers are

$b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$

defined through harmonic forms (Stepanov et al., 2013).

These meanings are related by homological language but not by a single common classification problem. A further source of ambiguity is that “Betti number classification” may refer either to classification by Betti invariants or to a downstream learning task that uses Betti-type descriptors. The supporting appendix for “GBNL: Graded Betti Number Learning of Complex Biological Data” documents a supervised prediction problem for binding free energy $\Delta G$, not a classification task, even though the title foregrounds graded Betti number learning (Zia et al., 27 Oct 2025).

2. Algebraic classification of graded and multigraded Betti tables

A central strand of the subject asks which graded Betti numbers can occur and how their support is constrained. For a monomial ideal $b_1=|E|-|V|+b_0,$0 with

$b_1=|E|-|V|+b_0,$1

one has the necessary condition

$b_1=|E|-|V|+b_0,$2

and, more sharply, a propagation rule: if

$b_1=|E|-|V|+b_0,$3

for $b_1=|E|-|V|+b_0,$4, then

$b_1=|E|-|V|+b_0,$5

This yields a precise forbidden region in the Betti table and is the main general classification statement for candidate nonzero bidegrees (Pour, 2015).

For multigraded modules of generic type, the classification becomes considerably more rigid. If $b_1=|E|-|V|+b_0,$6 is generic relative to a multigraded module $b_1=|E|-|V|+b_0,$7, then there is at most one homological degree $b_1=|E|-|V|+b_0,$8 for which $b_1=|E|-|V|+b_0,$9, and in that degree the value is the Crapo $b_1=\frac12|V|+1$0-invariant of a matroid minor $b_1=\frac12|V|+1$1: $b_1=\frac12|V|+1$2 For modules of generic type, this gives a complete combinatorial description of all multigraded Betti numbers (Charalambous et al., 2010).

The asymptotic version of the classification problem concerns powers and filtrations. For a homogeneous ideal $b_1=\frac12|V|+1$3 with generator degrees $b_1=\frac12|V|+1$4, the graded Betti numbers of $b_1=\frac12|V|+1$5 are eventually controlled by finitely many linear functions $b_1=\frac12|V|+1$6, with each slope $b_1=\frac12|V|+1$7 among the generator degrees, and by finitely many quasi-polynomials $b_1=\frac12|V|+1$8. Outside the strip bounded by the $b_1=\frac12|V|+1$9, the graded Tor groups vanish; inside each strip, and on each residue class modulo a period $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$0, the values are polynomial in $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$1 (Lamei et al., 2016). The same piecewise quasi-polynomial structure extends to $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$2-good filtrations $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$3, provided the filtration Rees algebra is finite over the ordinary Rees algebra (Lamei et al., 2016).

A negative boundary to any naïve global classification is provided by Bresinsky’s curves in $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$4. In that family one has

$$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$5

so all nonzero total Betti numbers are unbounded as $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$6 grows through even integers. This shows that even in embedding dimension $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$7, there is no uniform upper bound on total Betti numbers for these monomial curve ideals (Mehta et al., 2018).

3. Exact classification results in structured algebraic families

Several papers achieve full or partial Betti-number classification in rigid families where the resolution is governed by explicit combinatorics. For Kunz–Waldi semigroups $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$8, the determinantal subclass $$\beta_{i,j}(M)=\dim_K \operatorname{Tor}_i^S(K,M)_j,$$9 is characterized by arithmetic progressions in the parameters $j$0 and $j$1, equivalently by defining ideals generated by the $j$2 minors of a $j$3 matrix, or by pseudo-Frobenius numbers forming an arithmetic progression. In this subclass, and for every semigroup lying on the same face of the Kunz cone as such a determinantal example, the total Betti numbers are

$j$4

The paper is explicit that this is not yet a complete classification for all $j$5-semigroups; the full statement remains conjectural (González-Sánchez et al., 20 Mar 2025).

For the skeletons $j$6 of a special class of fat forests, the graded Betti table is restricted to two diagonals. The only nonzero graded Betti numbers are

$j$7

with explicit closed formulas in terms of the facet sizes $j$8, the number of facets $j$9, and the skeleton parameter $i$0. This also yields

$i$1

for $i$2, and a Cohen–Macaulay classification: the $i$3- and $i$4-skeletons are Cohen–Macaulay, while higher skeletons are generally not unless $i$5 (Fröberg, 25 Feb 2026).

For split graphs, the edge ring $i$6 has a particularly rigid Betti table: the only nonzero graded Betti numbers are $i$7 and $i$8 for $i$9. Moreover, the values depend only on the multiset of neighbor counts

$\beta_{i,\alpha}(L)$0

for the clique vertices $\beta_{i,\alpha}(L)$1. The same paper also determines exactly when the edge ring is Cohen–Macaulay: $\beta_{i,\alpha}(L)$2 (Fröberg, 2 Jun 2026).

Circulant graphs provide another family where explicit formulas are available. For the three families studied in “Graded Betti numbers of some circulant graphs,” the computations reduce to joins of complements of cycles, joins of cycles, or complete multipartite graphs. The first family has nonzero Betti numbers only on the diagonals $\beta_{i,\alpha}(L)$3 and $\beta_{i,\alpha}(L)$4; the second family can have multiple nonlinear diagonals governed by $\beta_{i,\alpha}(L)$5; and the third family has a completely linear resolution (Anand et al., 2020).

4. Graphs, cycle rank, and induced-subgraph classification

In graph theory, Betti number classification often means organization by the first Betti number $\beta_{i,\alpha}(L)$6 rather than by graded syzygies. For connected bipartite graphs with fixed first Betti number $\beta_{i,\alpha}(L)$7, every graph can be reduced by deleting leaves and contracting degree-$\beta_{i,\alpha}(L)$8 structures to a finite “basic graph” $\beta_{i,\alpha}(L)$9. The set $\alpha\in \mathbb Z^m$0 of such basic graphs is finite for fixed $\alpha\in \mathbb Z^m$1, and the generating function $\alpha\in \mathbb Z^m$2 decomposes as a finite sum over $\alpha\in \mathbb Z^m$3. This yields a genuine finite structural classification of connected bipartite graphs by fixed $\alpha\in \mathbb Z^m$4, together with PDE recurrences and asymptotic formulas (Hasui et al., 2022).

A more rigid topological-minor classification appears in the torus-embedding problem for cubic graphs. If $\alpha\in \mathbb Z^m$5 is a cubic graph with Betti number at most eight and does not embed into the torus, then $\alpha\in \mathbb Z^m$6 is one of exactly eleven pairwise non-isomorphic graphs

$\alpha\in \mathbb Z^m$7

and the actual cubic torus obstructions among these are precisely

$\alpha\in \mathbb Z^m$8

An important consequence is that there are no cubic torus obstructions of Betti number $\alpha\in \mathbb Z^m$9; the smallest ones occur exactly at Betti number $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$0 (Kramer, 1 Jul 2025).

For normal edge rings, the induced-subgraph approach gives a multigraded classification. In two-ear graphs and compact graphs of type $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$1 or $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$2, every nonzero multigraded Betti number is the top multigraded Betti number of some induced subgraph, and every nonzero value is $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$3. For compact graphs of type $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$4, this fails: some Betti numbers arise from second-top multigraded Betti numbers of type $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$5 induced subgraphs, and some values are $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$6. The final classification is

$b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$7

which isolates precisely where induced-subgraph top-support classification succeeds and where it requires second-top corrections (Wang et al., 2024).

5. Geometric and topological rigidity phenomena

In algebraic geometry, Betti-number classification often concerns either the Betti table of a projective embedding or the ordinary Betti numbers of a manifold or fibre. For graph curves arising from connected line arrangements, the quadratic strand is governed by a universal formula

$b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$8

so the full Betti table is determined once the cubic strand is known. This yields complete classifications for genus $b_r(M)=\dim H^r(M,\mathbb R)=\dim \ker \Delta,$9 and genus $\Delta G$0, and explicit formulas for trees of cycles in higher genus (Bruce et al., 2012).

For canonical curves, the classification problem focuses on the extremal linear Betti number. If $\Delta G$1 is a smooth non-maximally $\Delta G$2-gonal curve of genus $\Delta G$3, satisfying bpf-linear growth and general-position hypotheses for its minimal pencils, then

$\Delta G$4

where $\Delta G$5 is the number of minimal pencils. In the generic two-pencil case this becomes

$\Delta G$6

Here the value of the extremal Betti number classifies the curve by the number of minimal pencils under the stated transversality hypotheses (Kemeny, 2018).

For quasitoric manifolds with second Betti number $\Delta G$7, a different rigidity theorem holds: two such manifolds are homeomorphic if and only if their cohomology rings are isomorphic as graded rings. The condition $\Delta G$8 forces the orbit polytope to be a product of two simplices, and the classification reduces to explicit normal forms $\Delta G$9 and their specializations $b_1=|E|-|V|+b_0,$00 (Choi et al., 2010).

A related but distinct use of Betti numbers appears in complex polynomial maps $b_1=|E|-|V|+b_0,$01, where the relevant invariant is the top Betti defect

$b_1=|E|-|V|+b_0,$02

If $b_1=|E|-|V|+b_0,$03, then both the affine singular locus and the singular locus at infinity must be isolated. If

$b_1=|E|-|V|+b_0,$04

then singularities at infinity are still isolated, and the only non-isolated affine singularity allowed is a line with generic Morse transversal type, transverse to the hyperplane at infinity (Siersma et al., 2011).

6. Persistence, stochastic topology, and machine-learning uses

In applied topology, Betti-number classification is typically informative but not complete. For $b_1=|E|-|V|+b_0,$05-parameter persistence modules, bigraded Betti numbers record generators, first syzygies, and second syzygies in a minimal free resolution, but they do not classify modules up to isomorphism. A precise combinatorial formula nevertheless exists: if $b_1=|E|-|V|+b_0,$06 is the multiplicity of the interval $b_1=|E|-|V|+b_0,$07 in the barcode of the into-$b_1=|E|-|V|+b_0,$08 frame, and $b_1=|E|-|V|+b_0,$09 is the multiplicity of $b_1=|E|-|V|+b_0,$10 in the barcode of the $b_1=|E|-|V|+b_0,$11-outward frame, then

$b_1=|E|-|V|+b_0,$12

and $b_1=|E|-|V|+b_0,$13 is recovered from $b_1=|E|-|V|+b_0,$14, local dimensions, and $b_1=|E|-|V|+b_0,$15. The invariant is therefore computable from local zigzag barcodes, but it remains incomplete in the multiparameter setting (Moore, 2020).

For Gaussian random fields, the Betti numbers of excursion sets provide a threshold-resolved classification of topology. In three dimensions there are three relevant invariants: $b_1=|E|-|V|+b_0,$16 counts connected components, $b_1=|E|-|V|+b_0,$17 counts circular holes, and $b_1=|E|-|V|+b_0,$18 counts three-dimensional voids. Their alternating sum gives the genus: $b_1=|E|-|V|+b_0,$19 Numerically, $b_1=|E|-|V|+b_0,$20 dominates at high thresholds, $b_1=|E|-|V|+b_0,$21 near the median threshold, and $b_1=|E|-|V|+b_0,$22 at low thresholds. Unlike the Gaussian genus curve, whose shape is fixed, the amplitude and shape of the Betti number curves depend on the power-spectrum slope $b_1=|E|-|V|+b_0,$23 (Park et al., 2013).

Machine-learning applications based on graded Betti invariants are beginning to appear. “GBNL: Graded Betti Number Learning of Complex Biological Data” states that the method represents each nucleic acid sequence as a family of $b_1=|E|-|V|+b_0,$24-mer-specific sets and derives persistent graded Betti invariants from persistent commutative algebra, pairing them with transformer-based protein embeddings for protein–DNA/RNA binding prediction. At the same time, the supplied supporting appendix documents only instance-level experimental and predicted binding free energies $b_1=|E|-|V|+b_0,$25 on the datasets S186, S142, and S322, so the directly visible task is supervised regression rather than a classification problem (Zia et al., 27 Oct 2025).

Taken together, these works suggest a stable distinction. In some domains, Betti numbers classify objects exactly only under strong structural hypotheses, as with quasitoric manifolds having $b_1=|E|-|V|+b_0,$26 or generic multidegrees of multigraded modules (Choi et al., 2010, Charalambous et al., 2010). In others, they give a finite but partial organization, such as basic-graph reductions for fixed $b_1=|E|-|V|+b_0,$27, induced-subgraph top-support formulas, or asymptotic piecewise quasi-polynomial Betti tables (Hasui et al., 2022, Wang et al., 2024, Lamei et al., 2016). And in still others, they are deliberately incomplete but highly informative descriptors, as in multiparameter persistence, Gaussian excursion topology, and graded-Betti-based biological prediction (Moore, 2020, Park et al., 2013, Zia et al., 27 Oct 2025).