{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:11:06Z","timestamp":1787029866878,"version":"3.56.0"},"reference-count":53,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,10,13]],"date-time":"2017-10-13T00:00:00Z","timestamp":1507852800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We present the new skein invariants of classical links,     H [ H ]    ,     K [ K ]     and     D [ D ]    , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants     \u0398 d     related to the Yokonuma\u2013Hecke algebras and their 3-variable generalization    \u0398   , which generalizes the Homflypt polynomial.     H [ H ]     is the regular isotopy counterpart of    \u0398   . The invariants     K [ K ]     and     D [ D ]     are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant    \u0398    and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally.<\/jats:p>","DOI":"10.3390\/sym9100226","type":"journal-article","created":{"date-parts":[[2017,10,13]],"date-time":"2017-10-13T11:34:09Z","timestamp":1507894449000},"page":"226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Skein Invariants of Links and Their State Sum Models"],"prefix":"10.3390","volume":"9","author":[{"given":"Louis","family":"Kauffman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sofia","family":"Lambropoulou","sequence":"additional","affiliation":[{"name":"School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Athens, Greece"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,10,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kauffman, L.H., and Lambropoulou, S. (arXiv, 2017). New invariants of links and their state sum models, arXiv.","DOI":"10.3390\/sym9100226"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1090\/S0002-9947-1928-1501429-1","article-title":"Topological invariants of knots and links","volume":"30","author":"Alexander","year":"1928","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Conway, J.H. (1970). An enumeration of knots and links and some of their algebraic properties. Computational Problems in Abstract Algebra, Proceedings of the Conference Held at Oxford under the Auspices of the Science Research Council Atlas Computer Laboratory, Oxford, UK, 29 August\u20132 September 1967, Pergamon.","DOI":"10.1016\/B978-0-08-012975-4.50034-5"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1090\/S0273-0979-1985-15304-2","article-title":"A polynomial invariant for knots via von Neumann algebras","volume":"12","author":"Jones","year":"1985","journal-title":"Bull. Am. Math. Soc. (N.S.)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"335","DOI":"10.2307\/1971403","article-title":"Hecke algebra representations of braid groups and link polynomials","volume":"126","author":"Jones","year":"1987","journal-title":"Ann. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/0040-9383(87)90025-5","article-title":"A polynomial invariant of oriented links","volume":"26","author":"Lickorish","year":"1987","journal-title":"Topology"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1090\/S0273-0979-1985-15361-3","article-title":"A new polynomial invariant of knots and links","volume":"12","author":"Freyd","year":"1985","journal-title":"Bull. AMS"},{"key":"ref_8","first-page":"115","article-title":"Invariants of links of Conway type","volume":"4","author":"Przytycki","year":"1987","journal-title":"Kobe J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/0040-9383(87)90009-7","article-title":"State models and the Jones polynomial","volume":"26","author":"Kauffman","year":"1987","journal-title":"Topology"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1007\/BF01388747","article-title":"A polynomial invariant for unoriented knots and links","volume":"84","author":"Brandt","year":"1986","journal-title":"Invent. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1090\/S0002-9947-1990-0958895-7","article-title":"An invariant of regular isotopy","volume":"318","author":"Kauffman","year":"1990","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.aim.2012.10.011","article-title":"p\u2013adic framed braids II","volume":"234","author":"Juyumaya","year":"2013","journal-title":"Adv. Math."},{"key":"ref_13","unstructured":"Chlouveraki, M., Juyumaya, J., Karvounis, K., and Lambropoulou, S. (arXiv, 2016). Identifying the invariants for classical knots and links from the Yokonuma-Hecke algebra, arXiv."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1142\/S0218216504003020","article-title":"Markov trace on the Yokonuma-Hecke algebra","volume":"13","author":"Juyumaya","year":"2004","journal-title":"J. Knot Theory Ramif."},{"key":"ref_15","first-page":"825","article-title":"An adelic extension of the Jones polynomial","volume":"Volume 1","author":"Banagl","year":"2009","journal-title":"The Mathematics of Knots"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1350080","DOI":"10.1142\/S0218216513500806","article-title":"The Yokonuma-Hecke algebras and the Homflypt polynomial","volume":"22","author":"Chlouveraki","year":"2013","journal-title":"J. Knot Theory Ramif."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1641004","DOI":"10.1142\/S0218216516410042","article-title":"On the knot invariants from the Yokonuma\u2013Hecke algebras","volume":"25","author":"Chmutov","year":"2016","journal-title":"J. Knot Theory Ramif."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1641012","DOI":"10.1142\/S0218216516410121","article-title":"Enabling computations for link invariants coming from the Yokonuma-Hecke algebras","volume":"25","author":"Karvounis","year":"2016","journal-title":"J. Knot Theory Ramif."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.aim.2014.03.017","article-title":"Representation theory of the Yokonuma\u2013Hecke algebra","volume":"259","author":"Chlouveraki","year":"2014","journal-title":"Adv. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1641001","DOI":"10.1142\/S0218216516410017","article-title":"Tied Links","volume":"25","author":"Aicardi","year":"2016","journal-title":"J. Knot Theory Ramif."},{"key":"ref_21","unstructured":"Lambropoulou, S., Stefaneas, P., Theodorou, D., and Kauffman, L.H. (2015). Link invariants from the Yokonuma-Hecke algebras. Algebraic Modeling of Topological and Computational Structures and Applications, THALES Workshop. to appear November 2017."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1142\/S0218216509007324","article-title":"An invariant for singular knots","volume":"18","author":"Juyumaya","year":"2009","journal-title":"J. Knot Theory Ramif."},{"key":"ref_23","unstructured":"Juyumaya, J., and Lambropoulou, S. (arXiv, 2010). Modular framization of the BMW algebra, arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kauffman, L.H., and Manturov, V.O. (2014). On the framization of knot algebras. New Ideas in Low-Dimensional Topology, World Scientific.","DOI":"10.1142\/9348"},{"key":"ref_25","unstructured":"Goundaroulis, D. (2014). Framization of the Temperley-Lieb Algebra and Related Link Invariants. [Ph.D. Thesis, Department of Mathematics, National Technical University of Athens]."},{"key":"ref_26","first-page":"73","article-title":"The Yokonuma\u2013Temperley\u2013Lieb algebra","volume":"103","author":"Goundaroulis","year":"2014","journal-title":"Banach Center Pub."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s10468-014-9501-z","article-title":"Determination of the representations and a basis for the Yokonuma\u2013Temperley\u2013Lieb algebra","volume":"18","author":"Chlouveraki","year":"2015","journal-title":"Algebras Represent. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"299","DOI":"10.4310\/MRL.2017.v24.n2.a3","article-title":"Framization of the Temperley\u2013Lieb algebra","volume":"24","author":"Goundaroulis","year":"2017","journal-title":"Math. Res. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chlouveraki, M., and Pouchin, G. (arXiv, 2015). Representation theory and an isomorphism theorem for the Framisation of the Temperley\u2013Lieb algebra, arXiv.","DOI":"10.1007\/s00209-016-1751-5"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1007\/s00209-015-1598-1","article-title":"An isomorphism theorem for Yokonuma\u2013Hecke algebras and applications to link invariants","volume":"283","author":"Jacon","year":"2016","journal-title":"Math. Z."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1743005","DOI":"10.1142\/S0218216517430052","article-title":"Classical link invariants from the framizations of the Iwahori\u2013Hecke algebra and the Temperley\u2013Lieb algebra of type A","volume":"26","author":"Goundaroulis","year":"2017","journal-title":"J. Knot Theory Ramif."},{"key":"ref_32","unstructured":"Juyumaya, J. (arXiv, 2013). A partition Temperley\u2013Lieb algebra, arXiv."},{"key":"ref_33","unstructured":"Goundaroulis, D., and Lambropoulou, S. (arXiv, 2016). A new two-variable generalization of the Jones polynomial, arXiv."},{"key":"ref_34","unstructured":"Poulain d\u2019Andecy, L., and Wagner, E. (arXiv, 2016). The HOMFLYPT polynomials of sublinks and the Yokonuma\u2013Hecke algebras, arXiv."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"397","DOI":"10.17323\/1609-4514-2016-16-3-397-431","article-title":"Markov trace on the algebra of braids and ties","volume":"16","author":"Aicardi","year":"2016","journal-title":"Moscow Math. J."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Aicardi, F., and Juyumaya, J. (arXiv, 2016). Kauffman type invariants for tied links, arXiv.","DOI":"10.1142\/S0218216516410017"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4167","DOI":"10.1093\/imrn\/rnv257","article-title":"Markov trace on affine and cyclotomic Yokonuma\u2013Hecke algebras","volume":"14","author":"Chlouveraki","year":"2016","journal-title":"Int. Math. Res. Notices"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Flores, M., Juyumaya, J., and Lambropoulou, S. (2017). A Framization of the Hecke algebra of Type B. J. Pure Appl. Algebra.","DOI":"10.1016\/j.jpaa.2017.05.006"},{"key":"ref_39","first-page":"1","article-title":"State models for link polynomials","volume":"36","author":"Kauffman","year":"1990","journal-title":"Enseign. Math."},{"key":"ref_40","unstructured":"Kauffman, L.H. (2013). Knots and Physics, World Scientific Publishing Co. Pte. Ltd.. [4th ed.]."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1142\/S0218216501001050","article-title":"Links with trivial Jones polynomial","volume":"10","author":"Thistlethwaite","year":"2001","journal-title":"J. Knot Theory Ramif."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S0040-9383(02)00012-5","article-title":"Infinite families of links with trivial Jones polynomial","volume":"42","author":"Eliahou","year":"2003","journal-title":"Topology"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"311","DOI":"10.2140\/pjm.1989.137.311","article-title":"On knot invariants related to some statistical mechanical models","volume":"137","author":"Jones","year":"1989","journal-title":"Pac. J. Math."},{"key":"ref_44","unstructured":"Kauffman, L.H. (1983). Formal Knot Theory, Princeton University Press."},{"key":"ref_45","unstructured":"Baxter, R.J. (1982). Exactly Solved Models in Statistical Mechanics, Academic Press."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal statistics. I. A two-dimensional model with an order-disorder transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1090\/conm\/078\/975085","article-title":"Statistical Mechanics and the Jones Polynomial","volume":"Volume 78","author":"Kauffman","year":"1988","journal-title":"Braids"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1098\/rspa.1971.0067","article-title":"Relations between the \u201cpercolation\u201d and \u201ccolouring\u201d problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the \u201cpercolation\u201d problem","volume":"322","author":"Temperley","year":"1971","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Scheeler, M.W., Kleckner, D., Proment, D., Kindlmann, G.L., and Irvine, W.T.M. (arXiv, 2014). Helicity conservation by flow across scales in reconnecting vortex links and knots, arXiv.","DOI":"10.1073\/pnas.1407232111"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1017\/S0305004198002989","article-title":"Solving tangle equations arising in a DNA recombination model","volume":"126","author":"Ernst","year":"1999","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_51","unstructured":"Michieletto, D., Marenduzzo, D., and Turner, M.S. (arXiv, 2014). Topology Regulation during Replication of the Kinetoplast DNA, arXiv."},{"key":"ref_52","unstructured":"(2017, March 03). Kinetoplast. Available online: https:\/\/en.wikipedia.org\/wiki\/Kinetoplast."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Johri, S., Papic, Z., Schmitteckert, P., Bhatt, R.N., and Haldane, F.D.M. (arXiv, 2015). Probing the geometry of the Laughlin state, arXiv.","DOI":"10.1088\/1367-2630\/18\/2\/025011"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/226\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:47:11Z","timestamp":1760208431000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/9\/10\/226"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,10,13]]},"references-count":53,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2017,10]]}},"alternative-id":["sym9100226"],"URL":"https:\/\/doi.org\/10.3390\/sym9100226","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,10,13]]}}}