{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T08:16:57Z","timestamp":1783671417012,"version":"3.55.0"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T00:00:00Z","timestamp":1780876800000},"content-version":"vor","delay-in-days":7,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2026,6]]},"abstract":"Abstract<\/jats:title>\n \n Classical results of Cauchy [4] and Dehn [5] imply that the 1-skeleton of a convex simplicial polyhedron\n P<\/jats:italic>\n is rigid, i.e.\u00a0every continuous motion of the vertices of\n P<\/jats:italic>\n in\n \n \n $${\\mathbb R}^3$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n which preserves its edge lengths results in a polyhedron which is congruent to\n P<\/jats:italic>\n . This result was extended to convex simplicial polytopes in\n \n \n $${\\mathbb R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for all\n \n \n $$d\\ge 3$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by Whiteley [16], and to generic realisations of 1-skeletons of simplicial\n \n \n $$(d-1)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -manifolds in\n \n \n $${\\mathbb R}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n by Kalai [8] for\n \n \n $$d\\ge 4$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and Fogelsanger [6] for\n \n \n $$d\\ge 3$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We will generalise Kalai\u2019s result by showing that, for all\n \n \n $$d\\ge 4$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and any fixed\n \n \n $$1\\le k\\le d-3$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n \u2264<\/mml:mo>\n k<\/mml:mi>\n \u2264<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , every generic realisation of the\n k<\/jats:italic>\n -skeleton of a simplicial\n \n \n $$(d-1)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -manifold in\n \n \n $${\\mathbb R}^{d}$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is volume rigid, i.e.\u00a0every continuous motion of its vertices in\n \n \n $${\\mathbb R}^d$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n which preserves the volumes of its\n k<\/jats:italic>\n -faces results in a congruent realisation. In addition, we conjecture that our result remains true for\n \n \n $$k=d-2$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n =<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and verify this conjecture when\n \n \n $$d=4,5,6$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n =<\/mml:mo>\n 4<\/mml:mn>\n ,<\/mml:mo>\n 5<\/mml:mn>\n ,<\/mml:mo>\n 6<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s00493-026-00218-x","type":"journal-article","created":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T10:39:20Z","timestamp":1780915160000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Volume Rigidity of Simplicial Manifolds"],"prefix":"10.1007","volume":"46","author":[{"given":"James","family":"Cruickshank","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bill","family":"Jackson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shin-ichi","family":"Tanigawa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,8]]},"reference":[{"key":"218_CR1","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1090\/S0002-9947-1978-0511410-9","volume":"245","author":"L Asimow","year":"1978","unstructured":"Asimow, L., Roth, B.: The rigidity of graphs. 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