{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:28:41Z","timestamp":1772252921024,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,30]],"date-time":"2022-04-30T00:00:00Z","timestamp":1651276800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Quantum Gravity Research"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques\u2013Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers E\u02dc6 and D\u02dc4 contain fH. A surface birationally equivalent to a K3 surface is another compound of their character varieties.<\/jats:p>","DOI":"10.3390\/sym14050915","type":"journal-article","created":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T08:21:25Z","timestamp":1651652485000},"page":"915","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5739-546X","authenticated-orcid":false,"given":"Michel","family":"Planat","sequence":"first","affiliation":[{"name":"Institut FEMTO-ST CNRS UMR 6174, Universit\u00e9 de Bourgogne-Franche-Comt\u00e9, F-25044 Besan\u00e7on, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0637-1916","authenticated-orcid":false,"given":"Marcelo M.","family":"Amaral","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2490-877X","authenticated-orcid":false,"given":"Fang","family":"Fang","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4662-8592","authenticated-orcid":false,"given":"David","family":"Chester","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6953-8202","authenticated-orcid":false,"given":"Raymond","family":"Aschheim","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2938-3941","authenticated-orcid":false,"given":"Klee","family":"Irwin","sequence":"additional","affiliation":[{"name":"Quantum Gravity Research, Los Angeles, CA 90290, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"109","DOI":"10.2307\/2006973","article-title":"Varieties of group representations and splitting of 3-manifolds","volume":"117","author":"Culler","year":"1983","journal-title":"Ann. 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