Blueprint Bibliography🔗

Bibliography (34)

• B. Jackson and T. Jordán (2005). “Rigid two-dimensional frameworks with three collinear points”. Graphs Combin. 21, pp. 427–444.
  Cited from (1)

  • Document root
• B. Jackson and T. Jordán (2008). “Pin-collinear body-and-pin frameworks and the molecular conjecture”. Discrete Comput. Geom. 40, pp. 258–278.
  Cited from (1)

  • Document root
• B. Jackson, T. Jordán, and S. Villányi, 2026. “Rank contributions of vertices in rigidity matroids of clique covered graphs”. arXiv:2607.26266
  Cited from (5)

  • Document root - Cites Section 7.2
  • Document root
  • Chapter 1: Statement of the theorem, Section 1.4: Pin capacity and the partition condition - Cites Section 7.2
  • Chapter 1: Statement of the theorem, Section 1.1: The body-pin model - Cites Section 7.2
  • Chapter 1: Statement of the theorem, Section 1.5: The theorem
• C. St. J. A. Nash-Williams (1964). “Decomposition of finite graphs into forests”. J. London Math. Soc. 39.
  Cited from (1)

  • Chapter 8: Assembling the body-pin theorem, Section 8.2: Selecting a sparse subgraph
• Cs. Király and S.-i. Tanigawa, 2019. “Rigidity of body-bar-hinge frameworks”. In Handbook of Geometric Constraint Systems Principles, chapter 20. (Chapman & Hall/CRC, pp. 435–459)
  Cited from (6)

  • Document root
  • Document root
  • Chapter 9: Correspondence and audit, Section 9.2: Glossary
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.3: The grounded model
  • Chapter 1: Statement of the theorem, Section 1.4: Pin capacity and the partition condition - Cites reference Conjecture 5
  • Chapter 1: Statement of the theorem, Section 1.5: The theorem
• D. Eisenbud (1995). “Commutative Algebra with a View Toward Algebraic Geometry”. Graduate Texts in Mathematics, Springer. 150.
  Cited from (2)

  • Chapter 6: Degeneracy strata and the route not taken, Section 6.2: The codimension theorem for the strata
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.4: The dimension formula
• D. Zheng (20 August 2026). “Stress Degeneracy, Collinearity Flags, and Three-Dimensional Body–Pin Rigidity”. Research note. denzelzheng.com.
  Cited from (1)

  • Document root - Cites reference research note
• D. Zheng (September 2026). “Three-Dimensional Body–Pin Rigidity: a Lean 4 Formalization”. Lean 4 development. 1.3.0.
  Cited from (0)

  • No citation uses recorded.
• D. Zheng (August 2026). “Stress Degeneracy of Direction Complexes of (2, 2)-Sparse Graphs and Three-Dimensional Body–Pin Rigidity”. Preprint. doi:10.13140/RG.2.2.17830.28485.
  Cited from (69)

  • Document root
  • Chapter 8: Assembling the body-pin theorem
  • Chapter 8: Assembling the body-pin theorem, Section 8.4: Exceptional pin parameters, Lemma 8.4.1 - Cites reference Proposition 6.5
  • Chapter 8: Assembling the body-pin theorem, Section 8.2: Selecting a sparse subgraph, Lemma 8.2.1 - Cites Lemma 6.3
  • Chapter 8: Assembling the body-pin theorem, Section 8.5: The final assembly, Lemma 8.5.1 - Cites Section 6.4
  • Chapter 8: Assembling the body-pin theorem, Section 8.3: The orbit dimension drop, Lemma 8.3.1 - Cites Lemma 6.4
  • Chapter 8: Assembling the body-pin theorem, Section 8.1: The twist-equality partition, Definition 8.1.1 - Cites Section 6.1
  • Chapter 5: Collinearity flags
  • Chapter 5: Collinearity flags, Section 5.3: A global low-degree choice, Definition 5.3.1 - Cites Section 3.2
  • Chapter 5: Collinearity flags, Section 5.3: A global low-degree choice, Lemma 5.3.2 - Cites Lemma 3.4
  • Chapter 5: Collinearity flags, Section 5.1: Flags and their completions, Definition 5.1.1 - Cites reference Definition 3.1
  • Chapter 5: Collinearity flags, Section 5.1: Flags and their completions
  • Chapter 5: Collinearity flags, Section 5.1: Flags and their completions, Definition 5.1.2 - Cites reference Definition 3.2
  • Chapter 5: Collinearity flags, Section 5.6: Function-field branches and the semismallness budget, Theorem 5.6.1 - Cites Theorem 3.9
  • Chapter 5: Collinearity flags, Section 5.6: Function-field branches and the semismallness budget, Theorem 5.6.2 - Cites Theorem 1.2
  • Chapter 5: Collinearity flags, Section 5.4: Private support vertices and missing-edge pivots, Lemma 5.4.1 - Cites Lemma 3.5
  • Chapter 5: Collinearity flags, Section 5.4: Private support vertices and missing-edge pivots, Lemma 5.4.2 - Cites Lemma 3.6
  • Chapter 5: Collinearity flags, Section 5.2: The incidence forest, Lemma 5.2.1 - Cites reference Proposition 3.3
  • Chapter 5: Collinearity flags, Section 5.5: Two augmentation lemmas for the completion, Lemma 5.5.1 - Cites Lemma 3.7
  • Chapter 5: Collinearity flags, Section 5.5: Two augmentation lemmas for the completion, Lemma 5.5.2 - Cites Lemma 3.8
  • Chapter 9: Correspondence and audit
  • Chapter 9: Correspondence and audit, Section 9.4: Trust boundary
  • Chapter 9: Correspondence and audit, Section 9.4: Trust boundary - Cites Section A.2
  • Chapter 6: Degeneracy strata and the route not taken
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.2: The codimension theorem for the strata, Theorem 6.2.1 - Cites Theorem 4.2
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.2: The codimension theorem for the strata
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.2: The codimension theorem for the strata
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.1: The direction complex and its degeneracy loci, Definition 6.1.1 - Cites Equation 4.2–4.3
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.1: The direction complex and its degeneracy loci
  • Chapter 6: Degeneracy strata and the route not taken, Section 6.3: The grounded model, Lemma 6.3.1 - Cites Equation 4.7
  • Chapter 2: Necessity
  • Chapter 2: Necessity, Section 2.2: The counting argument, Lemma 2.2.1 - Cites Section 6.4
  • Chapter 2: Necessity, Section 2.1: Twists and the compatibility equation, Definition 2.1.1 - Cites Section 6.1
  • Chapter 2: Necessity, Section 2.1: Twists and the compatibility equation, Lemma 2.1.2 - Cites Lemma 6.1
  • Chapter 2: Necessity, Section 2.1: Twists and the compatibility equation, Lemma 2.1.3 - Cites Lemma 6.2
  • Chapter 3: Sparse graphs and addable edges
  • Chapter 3: Sparse graphs and addable edges, Section 3.4: A construction theorem with no paper counterpart
  • Chapter 3: Sparse graphs and addable edges, Section 3.3: An addable edge among three vertices, Lemma 3.3.1 - Cites Lemma 2.1
  • Chapter 3: Sparse graphs and addable edges, Section 3.1: Sparsity and tight sets, Definition 3.1.1 - Cites Equation 1.3
  • Chapter 3: Sparse graphs and addable edges, Section 3.2: Two consequences of supermodularity, Lemma 3.2.1 - Cites Section 2.1
  • Chapter 3: Sparse graphs and addable edges, Section 3.2: Two consequences of supermodularity, Lemma 3.2.2 - Cites Section 2.1
  • Chapter 1: Statement of the theorem
  • Chapter 1: Statement of the theorem, Section 1.4: Pin capacity and the partition condition, Definition 1.4.1 - Cites Equation 1.1
  • Chapter 1: Statement of the theorem, Section 1.4: Pin capacity and the partition condition
  • Chapter 1: Statement of the theorem, Section 1.4: Pin capacity and the partition condition, Definition 1.4.2 - Cites Equation 1.2, formally A.2
  • Chapter 1: Statement of the theorem, Section 1.1: The body-pin model, Definition 1.1.1 - Cites Section 1 and A.1
  • Chapter 1: Statement of the theorem, Section 1.1: The body-pin model, Definition 1.1.2 - Cites Section 1 and A.1
  • Chapter 1: Statement of the theorem, Section 1.5: The theorem, Theorem 1.5.1 - Cites Theorem 1.1
  • Chapter 1: Statement of the theorem, Section 1.5: The theorem, Theorem 1.5.2 - Cites Theorem A.1
  • Chapter 1: Statement of the theorem, Section 1.2: Two readings of generic rigidity, Definition 1.2.1 - Cites Equation 1.4
  • Chapter 1: Statement of the theorem, Section 1.2: Two readings of generic rigidity, Definition 1.2.2 - Cites Section 1 and A.1
  • Chapter 7: The Split-Klein isotropic-difference ideal
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.1: The Split-Klein form, Definition 7.1.1 - Cites Equation 5.1
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.3: The componentwise Witt shear, Lemma 7.3.1 - Cites Lemma 5.1
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.3: The componentwise Witt shear
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.4: The dimension formula, Lemma 7.4.1 - Cites Lemma 5.3
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.5: The height theorem, Theorem 7.5.1 - Cites Theorem 1.3
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.2: The ideal and the distinct locus, Definition 7.2.1 - Cites Equation 1.6 and 5.2
  • Chapter 7: The Split-Klein isotropic-difference ideal, Section 7.6: The ungrounded variety, Corollary 7.6.1 - Cites Corollary 5.4
  • Chapter 4: Vertex deletion and self-stress
  • Chapter 4: Vertex deletion and self-stress, Section 4.3: Certified response edges, Definition 4.3.1 - Cites Section 2.2
  • Chapter 4: Vertex deletion and self-stress, Section 4.3: Certified response edges
  • Chapter 4: Vertex deletion and self-stress, Section 4.2: Deleting one vertex, Definition 4.2.1 - Cites Section 2.2
  • Chapter 4: Vertex deletion and self-stress, Section 4.2: Deleting one vertex, Lemma 4.2.2 - Cites Equation 2.1–2.3
  • Chapter 4: Vertex deletion and self-stress, Section 4.2: Deleting one vertex, Definition 4.2.3 - Cites Equation 2.4–2.6
  • Chapter 4: Vertex deletion and self-stress, Section 4.5: Descent and the three neighbour rows, Lemma 4.5.1 - Cites Lemma 2.4
  • Chapter 4: Vertex deletion and self-stress, Section 4.5: Descent and the three neighbour rows, Lemma 4.5.2 - Cites Lemma 2.5
  • Chapter 4: Vertex deletion and self-stress, Section 4.1: Direction rows over a coefficient field, Definition 4.1.1 - Cites Section 2.2
  • Chapter 4: Vertex deletion and self-stress, Section 4.4: The local classification, Lemma 4.4.1 - Cites Lemma 2.3
• F. Doray, O. Karpenkov, and J. Schepers (2010). “Geometry of configuration spaces of tensegrities”. Discrete Comput. Geom. 43, pp. 436–466.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken
• G. Laman (1970). “On graphs and rigidity of plane skeletal structures”. J. Engrg. Math. 4, pp. 331–340.
  Cited from (1)

  • Document root
• J. Cruickshank, B. Jackson, T. Jordán, and S.-i. Tanigawa, 2026. “Rigidity of graphs and frameworks: A matroid theoretic approach”. In Surveys in Combinatorics 2026. (London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 189–230)
  Cited from (1)

  • Document root
• J. E. Graver, B. Servatius, and H. Servatius (1993). “Combinatorial Rigidity”. Graduate Studies in Mathematics, American Mathematical Society. 2.
  Cited from (1)

  • Document root
• J. Edmonds (1965). “Minimum partition of a matroid into independent subsets”. J. Res. Nat. Bur. Standards Sect. B. 69B, pp. 67–72.
  Cited from (1)

  • Chapter 8: Assembling the body-pin theorem, Section 8.2: Selecting a sparse subgraph
• K. Clinch, B. Jackson, and S.-i. Tanigawa (2022). “Abstract 3-rigidity and bivariate C¹₂-splines I: Whiteley's maximality conjecture”. Discrete Anal. 2022:2.
  Cited from (0)

  • No citation uses recorded.
• K. Clinch, B. Jackson, and S.-i. Tanigawa (2022). “Abstract 3-rigidity and bivariate C¹₂-splines II: Combinatorial characterization”. Discrete Anal. 2022:3.
  Cited from (1)

  • Document root
• L. Asimow and B. Roth (1978). “The rigidity of graphs”. Trans. Amer. Math. Soc. 245, pp. 279–289.
  Cited from (2)

  • Chapter 1: Statement of the theorem, Section 1.2: Two readings of generic rigidity
  • Chapter 1: Statement of the theorem, Section 1.2: Two readings of generic rigidity, Theorem 1.2.3 - Cites reference the theorem of §3, and Corollary 2
• L. Asimow and B. Roth (1979). “The rigidity of graphs, II”. J. Math. Anal. Appl. 68, pp. 171–190.
  Cited from (1)

  • Chapter 1: Statement of the theorem, Section 1.2: Two readings of generic rigidity
• L. Lovász and Y. Yemini (1982). “On generic rigidity in the plane”. SIAM J. Algebraic Discrete Methods. 3, pp. 91–98.
  Cited from (1)

  • Document root
• L. de Moura and S. Ullrich, 2021. “The Lean 4 theorem prover and programming language”. In Automated Deduction – CADE 28. (Lecture Notes in Computer Science 12699, Springer, pp. 625–635)
  Cited from (0)

  • No citation uses recorded.
• N. Katoh and S.-i. Tanigawa (2011). “A proof of the molecular conjecture”. Discrete Comput. Geom. 45, pp. 647–700.
  Cited from (1)

  • Document root
• N. L. White and W. Whiteley (1983). “The algebraic geometry of stresses in frameworks”. SIAM J. Algebraic Discrete Methods. 4, pp. 481–511.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken
• N. White and W. Whiteley (1987). “The algebraic geometry of motions of bar-and-body frameworks”. SIAM J. Algebraic Discrete Methods. 8, pp. 1–32.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken
• O. Karpenkov (2021). “The combinatorial geometry of stresses in frameworks”. Discrete Comput. Geom. 65, pp. 43–89.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken
• S. J. Gortler, A. D. Healy, and D. P. Thurston (2010). “Characterizing generic global rigidity”. Amer. J. Math. 132, pp. 897–939.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken
• T. Jordán, 2016. “Combinatorial rigidity: Graphs and matroids in the theory of rigid frameworks”. In Discrete Geometric Analysis. (MSJ Memoirs 34, Mathematical Society of Japan, pp. 33–112)
  Cited from (1)

  • Document root
• T.-S. Tay (1984). “Rigidity of multi-graphs. I. Linking rigid bodies in n-space”. J. Combin. Theory Ser. B. 36, pp. 95–112.
  Cited from (1)

  • Document root
• T.-S. Tay (1989). “Linking (n − 2)-dimensional panels in n-space II: (n − 2, 2)-frameworks and body and hinge structures”. Graphs Combin. 5, pp. 245–273.
  Cited from (1)

  • Document root
• The mathlib Community, 2020. “The Lean mathematical library”. In Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs. (ACM, pp. 367–381)
  Cited from (0)

  • No citation uses recorded.
• W. Bruns and H. J. Herzog (1998). “Cohen–Macaulay Rings, revised edition”. Cambridge Studies in Advanced Mathematics, Cambridge University Press. 39.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken, Section 6.2: The codimension theorem for the strata
• W. Bruns and U. Vetter (1988). “Determinantal Rings”. Lecture Notes in Mathematics, Springer. 1327.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken, Section 6.1: The direction complex and its degeneracy loci
• W. Fulton (1998). “Intersection Theory, second edition”. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer. 2.
  Cited from (1)

  • Chapter 6: Degeneracy strata and the route not taken, Section 6.1: The direction complex and its degeneracy loci
• W. Whiteley (1988). “The union of matroids and the rigidity of frameworks”. SIAM J. Discrete Math. 1, pp. 237–255.
  Cited from (1)

  • Document root
• W. Whiteley, 1996. “Some matroids from discrete applied geometry”. In Matroid Theory. (Contemporary Mathematics 197, American Mathematical Society, pp. 171–311)
  Cited from (1)

  • Document root