Banana Paths in Multidisciplinary Research

• Banana paths are families of routes joining two distinguished endpoints, appearing in mechanics, optics, quantum integrals, graph theory, and tropical geometry.
• In thin-shell mechanics and diffuse optical tomography, banana paths enable isometric deformations and optimal design rules for measurement sensitivity.
• In graph theory and topological data analysis, banana paths facilitate computational models of network invariants and dynamic persistence, while tropical KP uses them to encode soliton data.

“Banana paths” is used in the literature for several distinct objects: an isometric deformation path of banana-shaped thin shells; the banana-shaped source–detector sensitivity region in diffuse optical tomography; analytic integration, dispersion, and geodesic-distance paths for banana Feynman integrals; banana paths as multigraphs whose underlying simple graph is a path; dynamic path-based structures for persistent homology of time series; and path structures on banana curves and graphs in tropical KP theory (Couturier, 2016, Machida et al., 2022, Chen et al., 2024, Beougher et al., 3 Oct 2025, Ost et al., 2024, Abenda et al., 15 Dec 2025).

1. Terminological scope

In the cited literature, the phrase does not denote a single canonical object. It instead names families of trajectories, deformations, or combinatorial structures whose geometry is organized around two endpoints and multiple intermediate routes.

Domain	Meaning of “banana paths”	Representative paper
Thin-shell mechanics	Isometric deformation trajectories of banana-shaped shells	(Couturier, 2016)
Diffuse optics	Banana-shaped sensitivity region between source and detector	(Machida et al., 2022)
Feynman integrals	Dispersion-contour and iterated-integration paths for banana integrals	(Chen et al., 2024)
Graph theory	Banana trees or banana paths in multigraph theory	(Beougher et al., 3 Oct 2025)
Time-series topology	Dynamic path representation via banana trees	(Ost et al., 2024)
Tropical KP theory	Paths on banana graphs, Jacobians, and Delaunay polytopes	(Abenda et al., 15 Dec 2025)

This plurality is not accidental. In every case, the relevant structure is organized by a constrained family of routes between distinguished endpoints, but the endpoints may be shell apertures, optical probes, external legs of a Feynman graph, marked graph vertices, or nodes of a tropical curve.

2. Isometric shell kinematics and folded Goursat surfaces

In thin-shell mechanics, a banana path is the one-parameter family $h\mapsto S(u,v,h)$ of an explicit banana-shaped Goursat surface whose first fundamental form is independent of the deformation parameter $h$. The construction begins from a planar “banana-like” backbone $(U_1,U_2)$, an elliptic cross-section $(V_1,V_2)=(c\cos v,d\sin v)$, and the surface parametrization

$$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$

with metric coefficients $E,F,G$ independent of $h$. This makes the deformation exactly isometric, so the model captures bending-dominated shape change rather than stretching-dominated motion (Couturier, 2016).

The folded extension inserts a piecewise constant sign field $\epsilon(u)\in\{-1,1\}$ into the angular variable $\theta^\epsilon$. Discontinuities of $\epsilon$ create vertical curved folds along planes $h$0, and the resulting surfaces are $h$1 across fold lines. A fold energy is added to the bending energy, so the total energy takes the form

$h$2

The model is motivated by the seedpod of Acacia caven, where desiccation increases longitudinal curvature $h$3, decreases meridional curvature $h$4, and approximately preserves Gaussian curvature near a saddle point.

The central kinematic result is that a vertical actuation can be converted into qualitatively different horizontal responses. With no fold, increasing $h$5 opens the shell. With a fold placed close to the saddle or aperture, vertical closing produces horizontal closing. With the fold moved farther away, the fold influence weakens and opening is restored. The vertical deformation is measured by

$h$6

while the aperture response is measured by an opening area $h$7, positive for opening and negative for closing. The same formalism supports optimization under constant volume, and the reported conclusion is that the most elongated shells are optimal for opening ease, both geometrically and energetically.

3. Optical sensitivity regions in diffuse tomography

In diffuse optical tomography, “banana paths” denotes the banana-shaped region of highest measurement sensitivity between a source and a detector on the boundary of a semi-infinite half-space $h$8. With source $h$9 and detector $(U_1,U_2)$0, the first-order sensitivity of the detected signal to a point absorber at $(U_1,U_2)$1 is proportional to

$(U_1,U_2)$2

where $(U_1,U_2)$3 is the Green’s function of the diffusion equation. Its $(U_1,U_2)$4-$(U_1,U_2)$5 cross-section is symmetric about the midpoint, dives into the medium, reaches maximal depth near $(U_1,U_2)$6, and returns to the detector, producing the canonical banana shape (Machida et al., 2022).

The paper defines the banana depth physically rather than purely geometrically: it is the depth $(U_1,U_2)$7 at which a point absorber produces maximal reduction of detector fluence under the Born approximation. After nondimensionalization, the depth is

$(U_1,U_2)$8

where $(U_1,U_2)$9 is the smallest positive root of an integral equation $(V_1,V_2)=(c\cos v,d\sin v)$0. In the ideal limit of negligible absorption and zero boundary condition, this gives

$(V_1,V_2)=(c\cos v,d\sin v)$1

For typical tissue-like parameters and Robin boundary conditions with refractive-index mismatch, the reported rule of thumb is

$(V_1,V_2)=(c\cos v,d\sin v)$2

The same work uses stripe illumination, modeled as a periodic array of point sources. This superposes many individual banana-shaped sensitivity regions. Because stripe illumination contains both low and high spatial frequencies, it lies between point illumination and spatial-frequency-domain illumination: it retains deeper penetration than a single high-frequency sinusoidal pattern while still admitting Fourier decomposition. The paper uses the estimate $(V_1,V_2)=(c\cos v,d\sin v)$3 to choose stripe pitch $(V_1,V_2)=(c\cos v,d\sin v)$4, with effective source–detector distance

$(V_1,V_2)=(c\cos v,d\sin v)$5

This gives a practical design rule for depth targeting in DOT.

4. Analytic paths of banana Feynman integrals and modular periods

In perturbative quantum field theory, banana integrals are multi-loop two-point Feynman integrals with several propagators joining the same two external legs. For an $(V_1,V_2)=(c\cos v,d\sin v)$6-loop banana,

$(V_1,V_2)=(c\cos v,d\sin v)$7

with $(V_1,V_2)=(c\cos v,d\sin v)$8 and $(V_1,V_2)=(c\cos v,d\sin v)$9. In this context, banana paths are the branch-cut contours and nested integration paths appearing in dispersion relations and in parameterized discontinuities of banana integrals (p-DOBIs). The p-DOBIs organize discontinuities into nested integrals with Källén-function kernels, reduce to finite bases, and provide a direct route to Picard–Fuchs operators; the generalized dispersion relation adds threshold and UV subtractions through large and small contour contributions (Chen et al., 2024).

A complementary path language appears in the equal-mass three-loop banana graph in $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$0. There the graph is expressed as a double integral over auxiliary parameters $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$1, then as an outer integral over $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$2, with an inner iterated path in the $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$3-plane governed by kernels $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$4. The resulting representation organizes the three-loop banana as iterated integrals with algebraic kernels, suitable both for $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$5-expansion and for numerical evaluation above and below threshold (Bezuglov, 2021).

At zero external momentum, configuration space gives a different notion of banana path. A scalar $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$6-banana becomes an integral of a product of Euclidean propagators $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$7, hence a radial integral of products of Macdonald functions $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$8. For the 2-loop banana, the paper derives explicit formulas in arbitrary dimension and for arbitrary masses directly from the configuration-space representation, and it emphasizes that the relevant differential equations can be recovered from Bessel recurrences rather than from momentum-space IBP (Cacciatori et al., 2023).

The coordinate-space perspective persists on harmonic curved spaces. If the Green function depends only on geodesic distance $$S(u,v,h)=V_1(v) f(u,h)e_r(u,h)+\int_0^u U_3(t)\Gamma(t,h)\,dt+\left(\int_0^v\sqrt{V_2'(s)^2-hV_1'(s)^2}\,ds\right)e_z,$$9, then the $E,F,G$0-loop banana is simply $E,F,G$1. The paper shows that $E,F,G$2 satisfies a universal determinant differential equation built from a first-order radial operator $E,F,G$3 and an effective mass parameter $E,F,G$4,

$E,F,G$5

thereby turning banana diagrams into functions of geodesic distance on harmonic manifolds (Diakonov et al., 2024).

For maximal cuts of the three-loop banana, the analytic path reaches K3 geometry and automorphic forms. Depending on the mass pattern, the periods define ordinary modular forms or Hilbert, Siegel, or hermitian modular forms. Equal masses give elliptic modularity; three equal masses give a product-type elliptic situation on $E,F,G$6; pairwise-equal masses give Hilbert modular forms; two equal masses give Siegel modular forms on $E,F,G$7; and generic unequal masses give hermitian modular forms on a degree-2 hermitian half-space (Duhr, 21 Feb 2025).

5. Banana paths in graph theory, gonality, and induced subdivisions

In graph theory, a banana is the union of a nonempty set of positive-length paths with the same ends and otherwise disjoint interiors. A banana tree is obtained from a tree by replacing each edge by a banana, subject to orthogonality conditions between the bananas. This notion is used in the theory of induced subdivisions: if $E,F,G$8 is obtained from a tree by adding parallel edges, then $E,F,G$9 is pervasive, and the paper strengthens this by proving that such multigraphs are widespread. In bounded-clique graphs of sufficiently large chromatic number, induced subdivisions of banana trees must occur (Scott et al., 2017).

A different but related usage appears in chip-firing theory. A banana tree is a multigraph whose underlying simple graph is a tree, and a banana path is a banana tree whose underlying simple graph is a path. Such a path is specified by a sequence

$h$0

where $h$1 is the number of parallel edges between consecutive vertices. For $h$2, the paper defines a dynamic-programming function $h$3 and proves that $h$4 can be computed in time $h$5. It also proves an exact lcm formula for sufficiently long monoculture banana paths: $h$6 In the same setting, scramble number and screewidth coincide for banana trees, and banana paths are used to show that deleting a single edge can change gonality by an arbitrary prescribed amount while keeping the graph connected (Beougher et al., 3 Oct 2025).

Twice-marked banana graphs enter Hurwitz–Brill–Noether theory through two distinguished vertices $h$7. For genus $h$8, the banana graph is a theta graph, and the paper gives a sharp submodularity criterion: if $h$9 and $\epsilon(u)\in\{-1,1\}$0 lie in the interiors of distinct strands, every divisor is submodular. Evenly marked theta graphs—where the two marked points lie on distinct strands at the same rational position—have $\epsilon(u)\in\{-1,1\}$1-general transmission and become building blocks for new Brill–Noether general graphs. By contrast, almost all banana graphs of genus at least $\epsilon(u)\in\{-1,1\}$2 fail either by non-submodularity or by having too many inversions in the associated transmission permutations (Pflueger et al., 2022).

6. Dynamic time-series paths and persistent homology

In topological data analysis, a time series

$\epsilon(u)\in\{-1,1\}$3

is viewed as a piecewise-linear path $\epsilon(u)\in\{-1,1\}$4. The banana tree data structure maintains the extended $\epsilon(u)\in\{-1,1\}$5-dimensional persistence of this path under dynamic updates. Its geometric basis is a laminar family of double-panel windows associated with min–max persistence pairs. The up-tree stores the sublevel-set structure of $\epsilon(u)\in\{-1,1\}$6, the down-tree stores the corresponding structure for $\epsilon(u)\in\{-1,1\}$7, and each persistence pair is represented by a banana consisting of an in-trail and a mid-trail (Ost et al., 2024).

The update model includes value changes, path splitting, and path gluing. A value change decomposes into slides, anti-cancellations, cancellations, and interchanges of maxima or minima. The theoretical update bound is

$\epsilon(u)\in\{-1,1\}$8

where $\epsilon(u)\in\{-1,1\}$9 is the number of actual changes in the persistence diagram. The paper reports that for unbiased random walks the fraction of critical items is about $\theta^\epsilon$0, the average maximum nesting depth is about $\theta^\epsilon$1 for $\theta^\epsilon$2, and average in-trail and mid-trail lengths are both about $\theta^\epsilon$3. Under these conditions banana trees substantially outperform static recomputation.

The experimental comparison with Gudhi shows large speedups in the random-walk regime. For local value updates with $\theta^\epsilon$4, the median speedup at $\theta^\epsilon$5 is about $\theta^\epsilon$6. For cut and glue operations on unbiased random walks of length $\theta^\epsilon$7, average speedups are about $\theta^\epsilon$8. Real-world ECG, audio, and activity traces exhibit structural statistics close to unbiased random walks—short trails, logarithmic nesting depth, and small fractions of nodes on the longest trail—so the implementation is argued to have practical utility for dynamically evolving one-dimensional signals.

7. Banana curves, section classes, and tropical KP

In algebraic geometry, the banana threefold is a smooth proper Calabi–Yau threefold obtained by blowing up the relative diagonal in the fiber product of a rational elliptic surface with itself over $\theta^\epsilon$9. Its singular fibers contain a banana configuration

$\epsilon$0

a union of three $\epsilon$1s meeting in two points, with each $\epsilon$2 carrying normal bundle $\epsilon$3. The relevant curve classes form the rank-4 lattice

$\epsilon$4

where $\epsilon$5 is a section of the Abelian-surface fibration. The paper computes unweighted DT partition functions in classes involving both the banana configuration and the section, relates the connected PT theory to that of a rational elliptic surface, and extracts new unweighted GV invariants for classes $\epsilon$6 (Leigh, 2019).

In tropical KP theory, the banana graph $\epsilon$7 has two vertices and $\epsilon$8 parallel edges, so its genus is $\epsilon$9. The associated banana curve is a reducible rational curve obtained as a degeneration of a hyperelliptic curve. Its tropical Jacobian carries a Voronoi decomposition and dual Delaunay polytopes. For banana graphs, these Delaunay polytopes are combinatorially equivalent to hypersimplices $h$00, hence to the base polytopes of uniform matroids $h$01. The tropical theta divisor therefore canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions. The paper defines the Hirota variety of a banana graph as the parameter space of tau functions arising from the graph and gives an explicit parametrization that realizes those tau functions as multi-solitons (Abenda et al., 15 Dec 2025).

A plausible commonality is that these algebraic and tropical usages treat banana paths as curve classes or combinatorial trajectories in a Jacobian, rather than as literal geometric paths in ambient space. That commonality is structural rather than terminological: the central object is still a family of routes joining two distinguished components, now interpreted through periods, theta divisors, and soliton data.