×

zbMATH — the first resource for mathematics

Linking numbers in rational homology \(3\)-spheres, cyclic branched covers and infinite cyclic covers. (English) Zbl 1056.57007
Let \(K\cup K_1\cup\cdots\cup K_n\) be an oriented link in \(S^3\), \(lk(K, K_i)\) even. Then there is a surface \(F\cap K_i= \phi\), \(\partial F= K\). Using the Goeritz matrix of a basis of \(H_1(F)\), a linking-pairing \(\lambda_F(K_i, K_j)\), introduced in [Y. W. Lee, Proc. Am. Math. Soc. 126, No. 11, 3385–3392 (1998; Zbl 1001.57010)] is studied and proved to be invariant with respect to invariant changes of the basis and \(S^*\)-equivalence which is a non-orientable version of \(S\)-equivalence. If \(lk(K, K_i)= 0\), \(F\) is chosen as a Seifert surface.
The authors generalize the concept to rational homology spheres obtained by Dehn fillings with a framed link and give a formula for the difference of the linking numbers of a 2-component link resulting from different Dehn fillings: This generalizes a result by J. C. Cha and K.H. Ko [Topology 41, No. 6, 1161–1182 (2002; Zbl 1031.57020)]. The theory is then applied to finite cyclic coverings \(X_p\) of \(K\). The linking numbers in \(X_p\) of components of preimages \(K_1\), \(K_2\) (where \(K\cup K_1\cup K_2\) is a link in \(S^3\)) are expressed by \(\lambda_F(K_i, K_j)\) and \(lk(K_i, K_j)\). In the case of an infinite cyclic covering \(X_\infty\) an equivariant linking number is defined and similar results are proved: As a corollary the Kojima-Yamazaki \(\eta\)-function is given by a formula using the linking pairing. Contrary to the linking numbers defined \(\text{mod\,}\mathbb{Z}\) – or \(\text{mod\,}\mathbb{Z}[t,t^{-1}]\) in \(X_\infty\) – the linking pairing is absolute.
In the last chapter connections between the linking pairing and the signature of \(K\) (normal and Tristram-Levine) are proved.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M10 Covering spaces and low-dimensional topology
57M12 Low-dimensional topology of special (e.g., branched) coverings
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Selman Akbulut and Robion Kirby, Branched covers of surfaces in 4-manifolds, Math. Ann. 252 (1979/80), no. 2, 111 – 131. · Zbl 0421.57002 · doi:10.1007/BF01420118 · doi.org
[2] Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. · Zbl 0568.57001
[3] Jae Choon Cha and Ki Hyoung Ko, Signatures of links in rational homology spheres, Topology 41 (2002), no. 6, 1161 – 1182. · Zbl 1031.57020 · doi:10.1016/S0040-9383(01)00029-5 · doi.org
[4] U. Dahlmeier, Verkettungshomotopien in Mannigfaltigkeiten, Doktorarbeit Siegen 1994. · Zbl 0807.57003
[5] L. Goeritz, Knoten und quadratische Formen, Math. Z. 36 (1933), 647-654.
[6] C. McA. Gordon and R. A. Litherland, On the signature of a link, Invent. Math. 47 (1978), no. 1, 53 – 69. · Zbl 0391.57004 · doi:10.1007/BF01609479 · doi.org
[7] Jim Hoste, A formula for Casson’s invariant, Trans. Amer. Math. Soc. 297 (1986), no. 2, 547 – 562. · Zbl 0604.57008
[8] Gyo Taek Jin, On Kojima’s \?-function of links, Differential topology (Siegen, 1987) Lecture Notes in Math., vol. 1350, Springer, Berlin, 1988, pp. 14 – 30. · Zbl 0657.57002 · doi:10.1007/BFb0081466 · doi.org
[9] Uwe Kaiser, Link theory in manifolds, Lecture Notes in Mathematics, vol. 1669, Springer-Verlag, Berlin, 1997. · Zbl 0891.57008
[10] Louis H. Kauffman, Branched coverings, open books and knot periodicity, Topology 13 (1974), 143 – 160. · Zbl 0283.57011 · doi:10.1016/0040-9383(74)90005-6 · doi.org
[11] C. Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975), 141 – 160. · Zbl 0305.57016
[12] Sadayoshi Kojima and Masayuki Yamasaki, Some new invariants of links, Invent. Math. 54 (1979), no. 3, 213 – 228. · Zbl 0404.57004 · doi:10.1007/BF01390230 · doi.org
[13] R.H. Kyle, Branched covering spaces and the quadratic forms of links, Ann. of Math. 59 (1954), 539-548. · Zbl 0055.42103
[14] Youn W. Lee, A rational invariant for knot crossings, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3385 – 3392. · Zbl 1001.57010
[15] Youn W. Lee, Alexander polynomial for link crossings, Bull. Korean Math. Soc. 35 (1998), no. 2, 235 – 258. · Zbl 0907.57005
[16] Jerome Levine, Knot modules. I, Trans. Amer. Math. Soc. 229 (1977), 1 – 50. · Zbl 0653.57012
[17] Richard Mandelbaum and Boris Moishezon, Numerical invariants of links in 3-manifolds, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 285 – 304. · Zbl 0524.57025 · doi:10.1090/conm/020/718148 · doi.org
[18] Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387 – 422. · Zbl 0137.17903
[19] Dale Rolfsen, Knots and links, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. · Zbl 0339.55004
[20] Masahico Saito, On the unoriented Sato-Levine invariant, J. Knot Theory Ramifications 2 (1993), no. 3, 335 – 358. · Zbl 0809.57001 · doi:10.1142/S0218216593000192 · doi.org
[21] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig 1934. · JFM 60.0496.05
[22] M.V. Sokolov, Quantum invariants, skein modules, and periodicity of \(3\)-manifolds, Ph.D. Thesis, The George Washington University, Washington, D.C., 2000.
[23] A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251 – 264. · Zbl 0191.54703
[24] H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464 – 498. · Zbl 0108.18302 · doi:10.2307/1970369 · doi.org
[25] H. F. Trotter, On \?-equivalence of Seifert matrices, Invent. Math. 20 (1973), 173 – 207. · Zbl 0269.15009 · doi:10.1007/BF01394094 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.