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The Jones polynomial of ribbon links. (English) Zbl 1178.57002

The author studies the Jones polynomial from a topological point of view, with an eye toward finding results analogous to those known for the Alexander polynomial. For ribbon links, a Jones nullity is defined which features similarities to Murasugi’s nullity defined from the Seifert form. A family of link invariants generalizing the determinant is obtained. These invariants turn out to be of finite type with respect to band crossing changes, though not with respect to crossing changes. Certain congruences are found to hold for the generalized determinants.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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[1] J W Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275 · JFM 54.0603.03 · doi:10.2307/1989123
[2] D Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423 · Zbl 0898.57001 · doi:10.1016/0040-9383(95)93237-2
[3] D Bar-Natan, Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005) 1443 · Zbl 1084.57011 · doi:10.2140/gt.2005.9.1443
[4] J S Birman, X S Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993) 225 · Zbl 0812.57011 · doi:10.1007/BF01231287
[5] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter & Co. (1985) · Zbl 0568.57001
[6] A J Casson, C M Gordon, Cobordism of classical knots, Progr. Math. 62, Birkhäuser (1986) 181
[7] T D Cochran, Concordance invariance of coefficients of Conway’s link polynomial, Invent. Math. 82 (1985) 527 · Zbl 0589.57005 · doi:10.1007/BF01388868
[8] J H Conway, An enumeration of knots and links, and some of their algebraic properties, Pergamon (1970) 329 · Zbl 0202.54703
[9] M Eisermann, A geometric characterization of Vassiliev invariants, Trans. Amer. Math. Soc. 355 (2003) 4825 · Zbl 1033.57005 · doi:10.1090/S0002-9947-03-03117-9
[10] M Eisermann, Finite type invariants of surfaces in \(3\)-space, in preparation (2008)
[11] M Eisermann, C Lamm, A refined Jones polynomial for symmetric unions (2008) · Zbl 1246.57010
[12] D Erle, Quadratische Formen als Invarianten von Einbettungen der Kodimension \(2\), Topology 8 (1969) 99 · Zbl 0157.30901 · doi:10.1016/0040-9383(69)90002-0
[13] V Florens, On the Fox-Milnor theorem for the Alexander polynomial of links, Int. Math. Res. Not. (2004) 55 · Zbl 1080.57013 · doi:10.1155/S1073792804130894
[14] R H Fox, Some problems in knot theory, Prentice-Hall (1962) 168 · Zbl 1246.57011
[15] R H Fox, J W Milnor, Singularities of \(2\)-spheres in \(4\)-space and cobordism of knots, Osaka J. Math. 3 (1966) 257 · Zbl 0146.45501
[16] P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. \((\)N.S.\()\) 12 (1985) 239 · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3
[17] S Garoufalidis, Signatures of links and finite type invariants of cyclic branched covers, Contemp. Math. 231, Amer. Math. Soc. (1999) 87 · Zbl 0932.57005
[18] C M Gordon, Ribbon concordance of knots in the \(3\)-sphere, Math. Ann. 257 (1981) 157 · Zbl 0451.57001 · doi:10.1007/BF01458281
[19] M N Gusarov, A new form of the Conway-Jones polynomial of oriented links, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 193 (1991) 4, 161 · Zbl 0747.57005
[20] M Jacobsson, An invariant of link cobordisms from Khovanov homology, Algebr. Geom. Topol. 4 (2004) 1211 · Zbl 1072.57018 · doi:10.2140/agt.2004.4.1211
[21] V F R Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. \((\)N.S.\()\) 12 (1985) 103 · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2
[22] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987) · Zbl 0627.57002
[23] L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395 · Zbl 0622.57004 · doi:10.1016/0040-9383(87)90009-7
[24] L H Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990) 417 · Zbl 0763.57004 · doi:10.2307/2001315
[25] A Kawauchi, On the Alexander polynomials of cobordant links, Osaka J. Math. 15 (1978) 151 · Zbl 0401.57013
[26] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 · Zbl 0960.57005 · doi:10.1215/S0012-7094-00-10131-7
[27] M Khovanov, An invariant of tangle cobordisms, Trans. Amer. Math. Soc. 358 (2006) 315 · Zbl 1084.57021 · doi:10.1090/S0002-9947-05-03665-2
[28] R Kirby, editor, Problems in low-dimensional topology, AMS/IP Stud. Adv. Math. 2 (1997) · Zbl 0892.57013
[29] S Lang, Algebra, Graduate Texts in Mathematics 211, Springer (2002) · Zbl 0984.00001
[30] C Lescop, Global surgery formula for the Casson-Walker invariant, Annals of Mathematics Studies 140, Princeton University Press (1996) · Zbl 0949.57008
[31] J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229 · Zbl 0176.22101 · doi:10.1007/BF02564525
[32] W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer (1997) · Zbl 0886.57001
[33] C Livingston, A survey of classical knot concordance, Elsevier B. V., Amsterdam (2005) 319 · Zbl 1098.57006
[34] J W Milnor, Infinite cyclic coverings, Prindle, Weber & Schmidt, Boston (1968) 115 · Zbl 0179.52302
[35] Y Mizuma, Ribbon knots of 1-fusion, the Jones polynomial, and the Casson-Walker invariant, Rev. Mat. Complut. 18 (2005) 387 · Zbl 1084.57014 · doi:10.5209/rev_REMA.2005.v18.n2.16685
[36] Y Mizuma, An estimate of the ribbon number by the Jones polynomial, Osaka J. Math. 43 (2006) 365 · Zbl 1111.57007
[37] D Mullins, The generalized Casson invariant for \(2\)-fold branched covers of \(S^3\) and the Jones polynomial, Topology 32 (1993) 419 · Zbl 0784.57003 · doi:10.1016/0040-9383(93)90029-U
[38] H Murakami, T Ohtsuki, Finite type invariants of knots via their Seifert matrices, Asian J. Math. 5 (2001) 379 · Zbl 1015.57004
[39] H Murakami, T Ohtsuki, S Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. \((2)\) 44 (1998) 325 · Zbl 0958.57014
[40] K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387 · Zbl 0137.17903 · doi:10.2307/1994215
[41] G Polya, Induction and analogy in mathematics. Mathematics and plausible reasoning, vol. I, Princeton University Press (1954) · Zbl 0056.24101
[42] J H Przytycki, P Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988) 115 · Zbl 0655.57002
[43] J A Rasmussen, Khovanov homology and the slice genus, Invent. Math. (to appear) · Zbl 1211.57009 · doi:10.1007/s00222-010-0275-6
[44] D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish (1990) · Zbl 0854.57002
[45] A G Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969) 251 · Zbl 0191.54703
[46] V A Vassiliev, Cohomology of knot spaces, Adv. Soviet Math. 1, Amer. Math. Soc. (1990) 23 · Zbl 1015.57003
[47] O Viro, editor, Topology of manifolds and varieties, Advances in Soviet Mathematics 18, Amer. Math. Soc. (1994)
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