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The concordance classification of low crossing number knots. (English) Zbl 1322.57005

The subgroup \(\mathcal{F}\) of the classical knot concordance group \(\mathcal{C}\) generated by knots with at most eight crossings is shown to be isomorphic to \(\mathbb{Z}^{23}\oplus(\mathbb Z/2\mathbb Z)^7\), and its image in the algebraic knot concordance group \(\mathcal{G}\) is \(\mathbb{Z}^{18}\oplus{\mathbb Z/4\mathbb Z}\oplus(\mathbb Z/2\mathbb Z)^9\). (There are explicit bases.) The tools used are signatures and factorizations of the Alexander polynomial (as in Levine’s analysis of \(\mathcal{G}\)), Casson-Gordon invariants and twisted Alexander polynomials associated to characters of the homology of cyclic branched covers of low degree, implemented by computer calculations. The procedure is illustrated by working through the determination of the subgroup generated by the eight knots \(3_1, 7_2, _{10}, 8_{15}, 8_{18}\) and \(8_{20}\). An appendix summarizes work of the first author in her PhD thesis, in which the calculations are extended to determine almost completely the subgroup of \(\mathcal{C}\) generated by knots with at most nine crossings. In the ranges considered the TOP and DIFF concordance classifications agree.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Software:

KnotInfo
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Full Text: DOI arXiv

References:

[1] Casson, A. J.; Gordon, C. McA., Cobordism of classical knots. With an appendix by P. M. Gilmer. \`“A la recherche de la topologie perdue, Progr. Math. 62, 181-199 (1986), Birkh\'”auser Boston, Boston, MA
[2] [cha-livingston] J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, www.indiana.edu /\( \textasciitilde\) knotinfo, June 17, 2013. · Zbl 1049.57004
[3] [collins] J. Collins, On the concordance orders of knots, arxiv preprint: arXiv.org/abs/1206.0669 . · Zbl 1075.05603
[4] Cochran, Tim D.; Gompf, Robert E., Applications of Donaldson’s theorems to classical knot concordance, homology \(3\)-spheres and property \(P\), Topology, 27, 4, 495-512 (1988) · Zbl 0669.57003 · doi:10.1016/0040-9383(88)90028-6
[5] Cochran, T. D.; Lickorish, W. B. R., Unknotting information from \(4\)-manifolds, Trans. Amer. Math. Soc., 297, 1, 125-142 (1986) · Zbl 0643.57006 · doi:10.2307/2000460
[6] Cochran, Tim D.; Orr, Kent E.; Teichner, Peter, Structure in the classical knot concordance group, Comment. Math. Helv., 79, 1, 105-123 (2004) · Zbl 1061.57008 · doi:10.1007/s00014-001-0793-6
[7] Conway, J. H., An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 329-358 (1970), Pergamon, Oxford
[8] Fintushel, Ronald; Stern, Ronald J., Pseudofree orbifolds, Ann. of Math. (2), 122, 2, 335-364 (1985) · Zbl 0602.57013 · doi:10.2307/1971306
[9] Fox, Ralph H.; Milnor, John W., Singularities of \(2\)-spheres in \(4\)-space and cobordism of knots, Osaka J. Math., 3, 257-267 (1966) · Zbl 0146.45501
[10] Freedman, Michael H.; Quinn, Frank, Topology of 4-manifolds, Princeton Mathematical Series 39, viii+259 pp. (1990), Princeton University Press, Princeton, NJ · Zbl 0705.57001
[11] Gilmer, Patrick M., Slice knots in \(S^3\), Quart. J. Math. Oxford Ser. (2), 34, 135, 305-322 (1983) · Zbl 0542.57007 · doi:10.1093/qmath/34.3.305
[12] Greene, Joshua; Jabuka, Stanislav, The slice-ribbon conjecture for 3-stranded pretzel knots, Amer. J. Math., 133, 3, 555-580 (2011) · Zbl 1225.57006 · doi:10.1353/ajm.2011.0022
[13] Hedden, Matthew; Kirk, Paul, Instantons, concordance, and Whitehead doubling, J. Differential Geom., 91, 2, 281-319 (2012) · Zbl 1256.57006
[14] Herald, Chris; Kirk, Paul; Livingston, Charles, Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, Math. Z., 265, 4, 925-949 (2010) · Zbl 1210.57006 · doi:10.1007/s00209-009-0548-1
[15] Kirk, Paul; Livingston, Charles, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, 38, 3, 635-661 (1999) · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1
[16] Kirk, Paul; Livingston, Charles, Twisted knot polynomials: inversion, mutation and concordance, Topology, 38, 3, 663-671 (1999) · Zbl 0928.57006 · doi:10.1016/S0040-9383(98)00040-8
[17] Kirk, P.; Livingston, C., Concordance and mutation, Geom. Topol., 5, 831-883 (electronic) (2001) · Zbl 1002.57007 · doi:10.2140/gt.2001.5.831
[18] Levine, J., Invariants of knot cobordism, Invent. Math. {\bf 8} (1969), 98-110; addendum, ibid., 8, 355 pp. (1969) · Zbl 0179.52401
[19] Livingston, Charles, The algebraic concordance order of a knot, J. Knot Theory Ramifications, 19, 12, 1693-1711 (2010) · Zbl 1230.57008 · doi:10.1142/S0218216510008571
[20] Livingston, Charles; Naik, Swatee, Obstructing four-torsion in the classical knot concordance group, J. Differential Geom., 51, 1, 1-12 (1999) · Zbl 1025.57013
[21] Morita, Toshiyuki, Orders of knots in the algebraic knot cobordism group, Osaka J. Math., 25, 4, 859-864 (1988) · Zbl 0704.57014
[22] Long, D. D., Strongly plus-amphicheiral knots are algebraically slice, Math. Proc. Cambridge Philos. Soc., 95, 2, 309-312 (1984) · Zbl 0547.57006 · doi:10.1017/S0305004100061569
[23] Murasugi, Kunio, On a certain numerical invariant of link types, Trans. Amer. Math. Soc., 117, 387-422 (1965) · Zbl 0137.17903
[24] Rasmussen, Jacob, Khovanov homology and the slice genus, Invent. Math., 182, 2, 419-447 (2010) · Zbl 1211.57009 · doi:10.1007/s00222-010-0275-6
[25] Rudolph, Lee, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.), 29, 1, 51-59 (1993) · Zbl 0789.57004 · doi:10.1090/S0273-0979-1993-00397-5
[26] Tamulis, Andrius, Knots of ten or fewer crossings of algebraic order 2, J. Knot Theory Ramifications, 11, 2, 211-222 (2002) · Zbl 1003.57007 · doi:10.1142/S0218216502001585
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