{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:33Z","timestamp":1760144253436,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T00:00:00Z","timestamp":1710979200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201029"],"award-info":[{"award-number":["12201029"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Freeness occupies an important position in the study of hyperplane arrangements. In this paper, we conclude the freeness of three special classes of signed graphic arrangements based on the addition\u2013deletion theorem and Abe\u2019s free path theory.<\/jats:p>","DOI":"10.3390\/axioms13030208","type":"journal-article","created":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T06:35:30Z","timestamp":1711002930000},"page":"208","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Freeness of Signed Graphic Arrangements"],"prefix":"10.3390","volume":"13","author":[{"given":"Zhaoting","family":"Ju","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing University of Chemical Technology, 15 Beisanhuan East Road, Chaoyang District, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangfeng","family":"Jiang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Beijing University of Chemical Technology, 15 Beisanhuan East Road, Chaoyang District, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weili","family":"Guo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Beijing University of Chemical Technology, 15 Beisanhuan East Road, Chaoyang District, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1002\/cpa.3160290603","article-title":"Wave front evolution and equivariant Morse lemma","volume":"29","author":"Arnold","year":"1976","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1070\/RM1979v034n02ABEH002903","article-title":"Indices of singular points of 1-forms on a manifold with boundary, convolution of invariants of reflection groups, and singular projections of smooth surfaces","volume":"34","year":"1979","journal-title":"Russ. Math. Surv."},{"key":"ref_3","first-page":"265","article-title":"Theory of logarithmic differential forms and logarithmic vector fields","volume":"27","author":"Saito","year":"1980","journal-title":"J. Fac. Sci. Univ. Tokyo Sect. IA Math"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"389","DOI":"10.3792\/pjaa.56.389","article-title":"Free arrangements of hyperplanes and unitary reflection groups","volume":"56","author":"Terao","year":"1980","journal-title":"Proc. Jpn. Acad. Ser. A Math. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1023\/A:1022455716471","article-title":"Basic derivations for subarrangements of Coxeter arrangements","volume":"2","author":"Sagan","year":"1993","journal-title":"J. Algebr. Comb."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF02571719","article-title":"Free hyperplane arrangement between An\u22121 and Bn","volume":"215","author":"Edelman","year":"1994","journal-title":"Math. Z."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1007\/BF02944981","article-title":"Modular Elements of Geometric Lattices","volume":"1","author":"Stanley","year":"1971","journal-title":"Algebra Univ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1016\/j.aim.2007.04.019","article-title":"The characteristic polynomial of a multiarrangement","volume":"215","author":"Abe","year":"2007","journal-title":"Adv. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1080\/00029890.1981.11995201","article-title":"The geometry of root systems and signed graphs","volume":"88","author":"Zaslavsky","year":"1981","journal-title":"Am. Math. Mon."},{"key":"ref_10","unstructured":"Yoshinaga, M. (2004). Some characterizations of freeness of hyperplane arrangement. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1090\/S0002-9947-1990-0986703-7","article-title":"Matroid representations and free arrangements","volume":"320","author":"Ziegler","year":"1990","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_12","unstructured":"Bailey, G.D. (2024, February 27). Inductively Factored Signed-Graphic Arrangements of Hyperplanes. Available online: https:\/\/api.semanticscholar.org\/CorpusID:19031300."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s00373-018-1887-7","article-title":"On the Falk invariant of signed graphic arrangements","volume":"34","author":"Guo","year":"2018","journal-title":"Graphs Comb."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.disc.2018.09.029","article-title":"Signed graphs and the freeness of the Weyl subarrangements of type Bl","volume":"342","author":"Suyama","year":"2019","journal-title":"Discret. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3.10","DOI":"10.37236\/9341","article-title":"Freeness of Hyperplane Arrangements between Boolean Arrangements and Weyl Arrangements of Type Bl","volume":"27","author":"Torielli","year":"2020","journal-title":"Electron. J. Comb."},{"key":"ref_16","unstructured":"Orlik, P., and Terao, H. (2013). Arrangements of Hyperplanes, Springer Science Business Media."},{"key":"ref_17","first-page":"293","article-title":"Arrangements of hyperplanes and their freeness I, II","volume":"27","author":"Terao","year":"1980","journal-title":"Jpn. Fac. Sci. Univ. Tokyo"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0001-8708(86)90097-6","article-title":"Modular elements of lattices and topological fibration","volume":"62","author":"Terao","year":"1986","journal-title":"Adv. Math."},{"key":"ref_19","unstructured":"Abe, T., and Yamaguchi, T. (2023). Free paths of arrangements of hyperplanes. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0166-218X(82)90033-6","article-title":"Signed graphs","volume":"4","author":"Zaslavsky","year":"1982","journal-title":"Discret. Appl. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/208\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:17:13Z","timestamp":1760105833000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/3\/208"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,21]]},"references-count":20,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["axioms13030208"],"URL":"https:\/\/doi.org\/10.3390\/axioms13030208","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,3,21]]}}}