Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The concordance classification of low crossing number knots


Authors: Julia Collins, Paul Kirk and Charles Livingston
Journal: Proc. Amer. Math. Soc. 143 (2015), 4525-4536
MSC (2010): Primary 57M25; Secondary 57N70, 57Q45
DOI: https://doi.org/10.1090/proc/12587
Published electronically: April 29, 2015
MathSciNet review: 3373950
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present the complete classification of the subgroup of the classical knot concordance group generated by prime knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots.


References [Enhancements On Off] (What's this?)

  • [1] A. J. Casson and C. McA. Gordon, Cobordism of classical knots, With an appendix by P. M. Gilmer. À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 181-199. MR 900252
  • [2] J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, www.indiana.edu / $ \tilde {\ }$knotinfo, June 17, 2013.
  • [3] J. Collins, On the concordance orders of knots, arxiv preprint: arXiv.org/abs/1206.0669 .
  • [4] Tim D. Cochran and Robert E. Gompf, Applications of Donaldson's theorems to classical knot concordance, homology $ 3$-spheres and property $ P$, Topology 27 (1988), no. 4, 495-512. MR 976591 (90g:57020), https://doi.org/10.1016/0040-9383(88)90028-6
  • [5] T. D. Cochran and W. B. R. Lickorish, Unknotting information from $ 4$-manifolds, Trans. Amer. Math. Soc. 297 (1986), no. 1, 125-142. MR 849471 (87i:57003), https://doi.org/10.2307/2000460
  • [6] Tim D. Cochran, Kent E. Orr, and Peter Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no. 1, 105-123. MR 2031301 (2004k:57005), https://doi.org/10.1007/s00014-001-0793-6
  • [7] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, pp. 329-358. MR 0258014 (41 #2661)
  • [8] Ronald Fintushel and Ronald J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), no. 2, 335-364. MR 808222 (87a:57027), https://doi.org/10.2307/1971306
  • [9] Ralph H. Fox and John W. Milnor, Singularities of $ 2$-spheres in $ 4$-space and cobordism of knots, Osaka J. Math. 3 (1966), 257-267. MR 0211392 (35 #2273)
  • [10] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584 (94b:57021)
  • [11] Patrick M. Gilmer, Slice knots in $ S^{3}$, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 305-322. MR 711523 (85d:57004), https://doi.org/10.1093/qmath/34.3.305
  • [12] Joshua Greene and Stanislav Jabuka, The slice-ribbon conjecture for 3-stranded pretzel knots, Amer. J. Math. 133 (2011), no. 3, 555-580. MR 2808326 (2012e:57005), https://doi.org/10.1353/ajm.2011.0022
  • [13] Matthew Hedden and Paul Kirk, Instantons, concordance, and Whitehead doubling, J. Differential Geom. 91 (2012), no. 2, 281-319. MR 2971290
  • [14] Chris Herald, Paul Kirk, and Charles Livingston, Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation, Math. Z. 265 (2010), no. 4, 925-949. MR 2652542 (2011g:57006), https://doi.org/10.1007/s00209-009-0548-1
  • [15] Paul Kirk and Charles Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999), no. 3, 635-661. MR 1670420 (2000c:57010), https://doi.org/10.1016/S0040-9383(98)00039-1
  • [16] Paul Kirk and Charles Livingston, Twisted knot polynomials: inversion, mutation and concordance, Topology 38 (1999), no. 3, 663-671. MR 1670424 (2000c:57011), https://doi.org/10.1016/S0040-9383(98)00040-8
  • [17] P. Kirk and C. Livingston, Concordance and mutation, Geom. Topol. 5 (2001), 831-883 (electronic). MR 1871406 (2002j:57016), https://doi.org/10.2140/gt.2001.5.831
  • [18] J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98-110; addendum, ibid. 8 (1969), 355. MR 0253348 (40 #6563)
  • [19] Charles Livingston, The algebraic concordance order of a knot, J. Knot Theory Ramifications 19 (2010), no. 12, 1693-1711. MR 2755496 (2012e:57013), https://doi.org/10.1142/S0218216510008571
  • [20] Charles Livingston and Swatee Naik, Obstructing four-torsion in the classical knot concordance group, J. Differential Geom. 51 (1999), no. 1, 1-12. MR 1703602 (2000g:57009)
  • [21] Toshiyuki Morita, Orders of knots in the algebraic knot cobordism group, Osaka J. Math. 25 (1988), no. 4, 859-864. MR 983807 (90d:57028)
  • [22] D. D. Long, Strongly plus-amphicheiral knots are algebraically slice, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 2, 309-312. MR 735371 (85h:57007), https://doi.org/10.1017/S0305004100061569
  • [23] Kunio Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 0171275 (30 #1506)
  • [24] Jacob Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419-447. MR 2729272 (2011k:57020), https://doi.org/10.1007/s00222-010-0275-6
  • [25] Lee Rudolph, Quasipositivity as an obstruction to sliceness, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51-59. MR 1193540 (94d:57028), https://doi.org/10.1090/S0273-0979-1993-00397-5
  • [26] Andrius Tamulis, Knots of ten or fewer crossings of algebraic order 2, J. Knot Theory Ramifications 11 (2002), no. 2, 211-222. MR 1895371 (2003b:57012), https://doi.org/10.1142/S0218216502001585

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M25, 57N70, 57Q45

Retrieve articles in all journals with MSC (2010): 57M25, 57N70, 57Q45


Additional Information

Julia Collins
Affiliation: School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Scotland, EH9 3JZ
Email: Julia.Collins@ed.ac.uk

Paul Kirk
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: pkirk@indiana.edu

Charles Livingston
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: livingst@indiana.edu

DOI: https://doi.org/10.1090/proc/12587
Received by editor(s): November 19, 2013
Received by editor(s) in revised form: June 15, 2014
Published electronically: April 29, 2015
Additional Notes: This work was supported in part by the National Science Foundation under Grant 1007196, and by Simons Foundation Grants 278714 and 209082.
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society