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Liftings of surfaces in the plane

Oleg Karpenkov, Brigitte Servatius, Herman Servatius

TL;DR

This work generalizes Maxwell–Cremona liftings from planar frameworks to non-planar, surface-embedded frameworks by developing a combinatorial, monodromy-based lifting theory. It defines liftings along oriented face-paths on polygonal surfaces and proves homotopy-invariance of these lifts, then extends to non-planar frameworks with the monodromy-free condition. A key result shows that if the self-stress space has dimension $d>3b_1(S)$, nontrivial monodromy-free liftings exist, while for $b_1(S)=0$ the lifting problem becomes a bijection with piecewise-linear immersions, enabling explicit reconstruction. The paper also provides concrete examples, including a torus-like lifting, that demonstrate the range and nature of possible liftings in non-planar settings, with implications for rigidity theory and polyhedral geometry.

Abstract

In this note we provide a topological definition of Maxwell-Cremona liftings for non-planar frameworks of surfaces (both oriented and non-oriented). In the non-oriented case we give an estimate on the dimension of self-stresses, when the frameworks will posses a non-trivial lifting.

Liftings of surfaces in the plane

TL;DR

This work generalizes Maxwell–Cremona liftings from planar frameworks to non-planar, surface-embedded frameworks by developing a combinatorial, monodromy-based lifting theory. It defines liftings along oriented face-paths on polygonal surfaces and proves homotopy-invariance of these lifts, then extends to non-planar frameworks with the monodromy-free condition. A key result shows that if the self-stress space has dimension , nontrivial monodromy-free liftings exist, while for the lifting problem becomes a bijection with piecewise-linear immersions, enabling explicit reconstruction. The paper also provides concrete examples, including a torus-like lifting, that demonstrate the range and nature of possible liftings in non-planar settings, with implications for rigidity theory and polyhedral geometry.

Abstract

In this note we provide a topological definition of Maxwell-Cremona liftings for non-planar frameworks of surfaces (both oriented and non-oriented). In the non-oriented case we give an estimate on the dimension of self-stresses, when the frameworks will posses a non-trivial lifting.

Paper Structure

This paper contains 14 sections, 5 theorems, 16 equations, 3 figures.

Table of Contents

  1. Self-stresses and their liftings in the classical settings
  2. Self-stressed frameworks
  3. A classical notion of lifting for planar frameworks
  4. Lifts for oriented face-paths of planar surfaces
  5. Polygonal surfaces
  6. Oriented face-paths
  7. Lifts along oriented face-paths
  8. Homotopy invariance of lifts
  9. Liftings for non-planar frameworks
  10. Definition of lifting
  11. A few properties and remarks
  12. On monodromy-free liftings
  13. Liftings of surfaces with zero first Betti number
  14. Particular example

Key Result

Theorem 2.8

The lift $\tau_{\gamma,w}(f,f')$ does not depend on the choice of a face-path $\gamma$ from $f$ to $f'$ in the class of homotopic oriented face-paths.

Figures (3)

  • Figure 1: Lifting of a framework, corresponding to a torus.
  • Figure 2: Different types of lifting of a framework corresponding to a torus.
  • Figure 3: Stresses on a triangular prism. The dotted lines are unstressed. The plane framework lifts to an infinite triangular prism tube, corkscrewing out of the plane of the drawing. The unstressed dotted edges are all horizontal, at levels 32 units apart from one another. The stressed prism on the left lifts to a prism with two horizontal triangles, lifted $32$ units apart.

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • ...and 19 more