Spinning Families: Theories and Applications

• Spinning Families are a set of concepts where spin operations define diverse phenomena across algebraic combinatorics, geometric group theory, asteroid dynamics, topology, and high-energy physics.
• Key methodologies involve wreath product strategies, projection complex conditions, convex inversion techniques, and generating family constructions that yield classification theorems and free-product decompositions.
• Practical insights include solving generalized puzzles, modeling Yarkovsky effects in asteroid collisional families, constructing higher-dimensional topological objects, and inferring spin distributions in astrophysical contexts.

Searching arXiv for the primary paper and adjacent uses of “spinning families” across fields. The expression spinning families appears in several mathematically and physically distinct research programs. In algebraic combinatorics it denotes generalized spinning-switch puzzles modeled by wreath products; in geometric group theory it denotes equivariant collections of subgroups satisfying large-projection “spinning” conditions on projection complexes or cone-offs; in planetary science it refers to collisional asteroid families whose members’ spin states encode Yarkovsky and YORP evolution; and in topology and contact geometry it denotes spinning constructions that pass from lower-dimensional objects to higher-dimensional foams, Legendrians, and Lagrangian cobordisms (Kagey, 2022, Bestvina et al., 2020, Athanasopoulos et al., 2024, Carter et al., 2014). The common vocabulary comes from rotation, twist, or spin as an organizing operation, but the technical content is discipline-specific.

1. Wreath products and generalized spinning-switch puzzles

In the algebraic-combinatorial setting, the basic model starts with a finite group $G$ modeling a single switch and a finite group $H$ modeling the “spinning” action on a finite set $\Omega$ of switch positions. The puzzle is modeled by the wreath product

$$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$

with multiplication

$$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$

Here the base $G^\Omega$ records which move is applied at each corner, while the semidirect factor $H$ records the adversary’s rotation. A move is an element $(k,h)\in G\wr H$: the solver applies $k\in G^\Omega$ and then the adversary applies $h\in H$ (Kagey, 2022).

The central notion is a surjective strategy. A finite sequence $H$0 in the base $H$1 is called a surjective strategy if, for every adversarial choice $H$2, the partial products

$H$3

run through every element of $H$4. Equivalently, no matter the hidden start-state in $H$5, one of the partial products brings the system to the identity.

The main classification theorem in the abelian case states that if $H$6 is finite abelian and $H$7 acts faithfully on $H$8, then

$H$9

A second theorem states that if $\Omega$0 is a $\Omega$1-group of order $\Omega$2 and $\Omega$3 is any finite $\Omega$4-group acting faithfully, then $\Omega$5 admits a winning strategy. The proof uses induction on a proper normal $\Omega$6-subgroup $\Omega$7, together with an interleaving construction that combines strategies for $\Omega$8 and $\Omega$9 (Kagey, 2022).

The explicit constructions mirror the algebra. In the abelian case $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$0, one identifies $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$1 and constructs a sequence of base-moves whose partial sums sweep through the entire vector space. In the general $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$2-group case one fixes a chief series

$$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$3

with $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$4, and interleaves quotient and kernel strategies. The paper also gives a nonabelian example with two interchangeable copies of the Monster group $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$5: with $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$6 swapping the two copies, a surjective strategy is obtained because $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$7 is generated by involutions and the construction forces every pair $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$8 (Kagey, 2022).

This line of work recasts a class of recreational puzzles as a solvability problem for wreath products. The paper closes with open questions on full classification, higher-order generators, palindromic strategies, quasigroup switches, expected turns under random play, and bounds on minimal strategy length.

2. Projection complexes, rotating families, and quotient geometry

In geometric group theory, a spinning family is defined relative to a projection complex. Let $$G\wr H \;=\; G^\Omega\rtimes H, \qquad G^\Omega = \prod_{\omega\in\Omega}G_\omega,\;G_\omega\cong G,$$9 be a projection complex with vertex set $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$0, and let $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$1 act by isometries on $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$2, preserving the projection data. For each vertex $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$3 choose a subgroup $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$4. Then $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$5 is an $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$6-spinning family if it satisfies equivariance,

$$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$7

and the spinning condition,

$$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$8

The principal theorem states that there exists $$(k,h)\cdot(k',h') =\bigl(k\,(h\cdot k'),\,hh'\bigr), \quad h\cdot(g_\omega)=(\,g_{h^{-1}\omega}\,).$$9 such that, when $G^\Omega$0, the normal closure

$G^\Omega$1

is isomorphic to a free product

$G^\Omega$2

for some subset of representatives $G^\Omega$3; moreover every nontrivial element of $G^\Omega$4 is either conjugate into some $G^\Omega$5 or acts loxodromically on $G^\Omega$6 (Bestvina et al., 2020).

A related but more elaborate framework uses rotating families and composite rotating families on several projection complexes simultaneously. Here one works with a disjoint union

$G^\Omega$7

equipped with projection data and active sets, and with infinite subgroups

$G^\Omega$8

satisfying equivariance, large-rotation conditions, and commutation for disjoint support. Dahmani’s “plate-spinning” construction organizes these data via composite windmills and successive unfoldings through the coordinates. The resulting subgroup

$G^\Omega$9

admits a partially commutative presentation whose only relators are commutators

$H$0

whenever the relevant supports do not interact. In the mapping-class-group application, the normal closure of a sufficiently large power of a Dehn twist has a presentation whose relators consist only of commutators between twists of disjoint support (Dahmani, 2017).

A recent extension places spinning families in the setting of hyperbolic and hierarchically hyperbolic quotients. Let $H$1 be a Gromov-hyperbolic graph, let $H$2 be a $H$3-invariant collection of uniformly $H$4-quasiconvex subgraphs, and let $H$5 be the electrification obtained by coning off each $H$6. With cone vertex $H$7 and extended projections $H$8, an equivariant family $H$9 is $(k,h)\in G\wr H$0-spinning if for every $(k,h)\in G\wr H$1, every $(k,h)\in G\wr H$2 in $(k,h)\in G\wr H$3, and every nontrivial $(k,h)\in G\wr H$4,

$(k,h)\in G\wr H$5

The quotient graph $(k,h)\in G\wr H$6 is hyperbolic when $(k,h)\in G\wr H$7 is sufficiently large, and the same techniques show that random quotients of hierarchically hyperbolic groups are again hierarchically hyperbolic asymptotically almost surely (Abbott et al., 22 Jul 2025).

Across these works, spinning families function as a small-cancellation-like mechanism encoded in projection data rather than in a relator set. The resulting algebraic outputs are free-product decompositions, partial-commutativity theorems, Greendlinger-type shortening lemmas, and stability of hyperbolic or hierarchically hyperbolic quotient structures.

3. Spin-state asymmetry in asteroid collisional families

In asteroid science, the relevant “families” are collisional families: fragments of a disrupted parent body that initially cluster in proper orbital element space and subsequently disperse under thermal forces. The key theoretical relation is the Yarkovsky drift law

$(k,h)\in G\wr H$8

with prograde rotators moving outward and retrograde rotators moving inward. In the $(k,h)\in G\wr H$9 plane, the fragments define a V-shaped envelope

$k\in G^\Omega$0

where the slope encodes family age. Spin-state measurements therefore provide an independent test of family membership: one expects retrograde excess on the inward wing and prograde excess on the outward wing (Athanasopoulos et al., 2024).

This pattern is observed for the X-complex Athor and Zita families in the inner main belt. For Athor, the family contains 60% of retrograde asteroids on the inward side and 76% of prograde asteroids on the outward side; for Zita, the inward side contains 80% of retrograde asteroids, while the outward side has equal numbers of prograde and retrograde rotators, although kernel density estimation shows a clear prograde peak on the outward side. The analysis combines dense and sparse photometry, lightcurve construction, and convex inversion to estimate sidereal period, spin axis, and convex shape (Athanasopoulos et al., 2024).

A related study of a 4 Gyr collisional family in the inner main belt used photometric observations, literature light curves, and sparse-in-time photometry to determine spin poles for 55 candidate members. After rejecting 9 interlopers via albedo and spectral criteria, 46 confirmed family members remained, of which 31 are retrograde and 15 prograde. Under the null hypothesis of equal prograde and retrograde probability, the one-sided binomial p-value is

$k\in G^\Omega$1

The predominance of retrograde spins on the inward wing was therefore taken as corroborating evidence of common origin and long-term Yarkovsky-driven spreading (Athanasopoulos et al., 2022).

Methodologically, these studies rely on convex inversion with a scattering law combining Lommel-Seeliger and Lambert terms, followed by classification of the spin sense from the ecliptic latitude $k\in G^\Omega$2: prograde for $k\in G^\Omega$3, retrograde for $k\in G^\Omega$4. The family-scale inference does not require exact reconstruction of the original collision; rather, it uses the statistical persistence of the spin–orbit correlation over Gyr timescales. This suggests that spin-state analysis can validate very old and diffuse families that are difficult to isolate by orbital clustering alone.

4. Synthetic families, TESS rotational statistics, and normalized YORP evolution

A second asteroid-science use of the expression concerns synthetic families produced in impact simulations with rotating parent bodies. In SPH/N-body calculations of collisions into monolithic targets with diameters $k\in G^\Omega$5 and $k\in G^\Omega$6, the parent-body spin modifies fragmentation, reaccumulation, and the spin distribution of the fragments. The mass ejection relation is written

$k\in G^\Omega$7

with $k\in G^\Omega$8 for slow rotators and $k\in G^\Omega$9 reaching up to $h\in H$0 for near-critical spins in oblique cratering events. Although individual impacts can accelerate or decelerate the target, the deceleration prevails on average, so collisions cause a systematic spin-down of asteroid population (Ševeček et al., 2019).

Observationally, TESS S1–S13 light curves were used to study 2859 family asteroids in 16 Main Belt families. The distributions of rotation periods and amplitudes are family-specific in some groups, including the Hungaria, Maria, Juno, Eos, Eucharis, and Alauda families. Older families tend to contain a larger fraction of more spheroidal, low-amplitude asteroids. For eight families with firm ages, the log-linear regression is

$h\in H$1

The study also reports that very fast rotators are close to spherical, minor planets rotating slower than $h\in H$2 hour are also more spherical than asteroids in the 4--8 hour period range, and the Flora and Maria families show statistically significant core-versus-outskirts differences in period distributions (Szabó et al., 2022).

A still broader synthesis is provided by a 2026 analysis of 8739 asteroids with spin period measurements and 3794 asteroids with obliquity determinations across 28 asteroid families spanning ages from 14 Myr to 3 Gyr. The paper introduces the dimensionless time

$h\in H$3

and studies two observables. The slow-rotator fraction $h\in H$4, defined using $h\in H$5 h, increases steeply and saturates at

$h\in H$6

around a breakpoint

$h\in H$7

The polarization fraction $h\in H$8, which measures whether the spin-pole sign matches the expected side of the family V-shape, rises to a maximum of $h\in H$9 near $H$00 and then decays toward the random limit $H$01 for $H$02 (Bertinelli et al., 19 Jan 2026).

Taken together, these results show that asteroid-family “spinning” can refer both to the spin of the disrupted parent body and to the post-collisional spin evolution of family members. The first controls fragment production and angular-momentum drainage; the second controls the long-term observables used for family identification and age diagnostics.

5. Spinning constructions in topology, Legendrian theory, and Lagrangian cobordism

In low-dimensional and symplectic topology, spinning is a construction rather than a statistical descriptor. For knotted trivalent graphs (KTGs), the starting object is a spatial embedding

$H$03

of a trivalent graph. The relevant higher-dimensional objects are 2-dimensional foams, spaces locally modeled on the cone on the 1-skeleton of the 3-simplex. The classical Zeeman theorem implies that the $H$04-twist spin of any classical knot is an unknotted $H$05-sphere, but the graph case is different: there exists a knotted $H$06-curve $H$07 and an edge $H$08 such that its $H$09-twist spin

$H$10

is a knotted foam (Carter et al., 2014).

The same paper proves a deletion criterion. If $H$11 is unknotted, then the $H$12-twist spin $H$13 is an unknotted foam. Consequently, if $H$14 is unknotted or almost trivial, then for each edge $H$15 the $H$16-twist spin is unknotted. Kinoshita’s $H$17-curve is almost trivial, so every $H$18-twist spin of it is unknotted; by contrast, a $H$19-curve whose constituent knots are all granny knots has the property that every $H$20-twist spin is knotted (Carter et al., 2014).

A related but distinct spinning construction appears in the theory of generating families for Legendrian submanifolds. Let

$H$21

be a linear-at-infinity generating family for a Legendrian in $H$22, with $H$23 in the spinning construction. The function is called spinnable if it extends smoothly across the reflecting hyperplane, equivalently

$H$24

Introducing polar coordinates $H$25 on the $H$26-plane, the spun generating family is

$H$27

with fiber-critical set

$H$28

This $H$29 generates an embedded spun Legendrian $H$30. If one begins with a spinnable, gf-compatible exact Lagrangian cobordism, the same construction produces an embedded, gf-compatible exact Lagrangian cobordism between the spun Legendrians (Bourgeois et al., 2014).

In both settings the word “spinning” is literal: it denotes a controlled geometric operation, usually an $H$31-rotation, that creates higher-dimensional objects while preserving specific regularity or compatibility conditions.

6. Further specialized usages in high-energy theory, compact objects, quantum droplets, and binary black holes

A further use of spin-family language appears in a six-dimensional toy model for the spin-charge-family theory. In $H$32, with a zweibein that curves an infinite disc into an almost $H$33 and with two kinds of spin-connection fields $H$34 and $H$35, the theory contains

$H$36

independent families. After the $H$37 split, the equations of motion separate into two decoupled pairs of families, $H$38 and $H$39. Only for the fine-tuned choice

$H$40

do normalizable zero-modes survive, reproducing a single-chiral 4D Weyl spinor (Lukman et al., 2012).

In compact-star astrophysics, the relevant notion is not a spinning construction but two coexisting families of compact stars, namely neutron stars and strange stars. Fully general relativistic computations for uniformly rotating stars show that neutron-star $H$41 strange-star conversion, under baryon-number and angular-momentum conservation, causes a simultaneous spin-up and decrease in gravitational mass: $H$42 This produces a new evolutionary channel in the $H$43–$H$44 plane, distinct from ordinary accretion-driven spin-up or electromagnetic spin-down (Bhattacharyya et al., 2017).

In nonlinear quantum many-body theory, families of spinning quantum droplets arise in a two-dimensional amended Gross–Pitaevskii equation with Lee–Huang–Yang corrections and a $H$45-symmetric potential. In the unbroken-$H$46 region, stationary multipolar and vortex-droplet families bifurcate from linear modes; in the spinning regime, droplets become asymmetric above a critical rotation frequency $H$47, but many remain stable. For the monopole family at $H$48, the paper reports

$H$49

The same work describes essentially elastic collisions for both spinning and nonspinning droplets under the chosen trap (Song et al., 2023).

Finally, in gravitational-wave population inference, “spinning” and “nonspinning” binary-black-hole populations are disentangled by spin sorting. Defining

$H$50

one compares purely nonspinning, singly-spinning, and fully spinning synthetic populations under an IID Beta spin-magnitude model. The study concludes that current observations are inconsistent with a fully nonspinning population, but could be explained by a population with only one spinning black hole per binary or a population with up to 80% nonspinning sources (Szemraj et al., 31 Jul 2025).

These specialized usages do not share a common formalism. What they share is a structural role for spin as a family-defining parameter: a family index in higher-dimensional fermion models, an evolutionary discriminator in compact objects and binary black holes, or a dynamical control parameter in non-Hermitian quantum droplets.