{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T03:12:36Z","timestamp":1761707556077},"reference-count":9,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Knot Theory Ramifications"],"published-print":{"date-parts":[[2003,2]]},"abstract":"<jats:p> In [2] Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a satisfying the skein relations: [Formula: see text] and is defined in terms of a state-sum and the Dubrovnik polynomial for links. <\/jats:p><jats:p> In [4] it is proved, in the case B = A<jats:sup>- 1<\/jats:sup> and a = A, that for a planar garph G we have [G] = 2<jats:sup>c - 1<\/jats:sup> (- A - A<jats:sup>- 1<\/jats:sup>)<jats:sup>v<\/jats:sup>, where c is the number of connected components of G and v is the number of vertices of G. <\/jats:p><jats:p> In this paper we will show how we can calculate the polynomial, with the variables B = A<jats:sup>- 1<\/jats:sup> and a = A, without resorting to the skein relation. <\/jats:p>","DOI":"10.1142\/s0218216503002317","type":"journal-article","created":{"date-parts":[[2003,2,5]],"date-time":"2003-02-05T06:21:29Z","timestamp":1044426089000},"page":"67-78","source":"Crossref","is-referenced-by-count":2,"title":["Topological notions for Kauffman and Vogel's polynomial"],"prefix":"10.1142","volume":"12","author":[{"given":"Rui Pedro","family":"Carpentier","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica and Centro  de Matem\u00e1tica Aplicada, Instituto Superior T\u00e9cnico, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1989-0946218-0"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218216592000069"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0958895-7"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0218216500000578"},{"key":"rf5","volume-title":"The Four-Color Theorem: Assaults and Conquest","author":"Saaty T. L.","year":"1977"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1142\/1116"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190130503"},{"key":"rf8","volume-title":"A Survey of Knot Theory","author":"Kawauchi A.","year":"1996"},{"key":"rf9","volume-title":"Knots, links, braids and 3-manifolds. An introduction to the new invariants in low-dimensional topology","author":"Prasolov V. V.","year":"1996"}],"container-title":["Journal of Knot Theory and Its Ramifications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218216503002317","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T03:37:13Z","timestamp":1565149033000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218216503002317"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,2]]},"references-count":9,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2003,2]]}},"alternative-id":["10.1142\/S0218216503002317"],"URL":"https:\/\/doi.org\/10.1142\/s0218216503002317","relation":{},"ISSN":["0218-2165","1793-6527"],"issn-type":[{"value":"0218-2165","type":"print"},{"value":"1793-6527","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,2]]}}}