Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Polynomial splittings of metabelian von Neumann rho-invariants of knots

Author(s): Se-Goo Kim; Taehee Kim
Journal: Proc. Amer. Math. Soc. 136 (2008), 4079-4087.
MSC (2000): Primary 57M25; Secondary 57N70
Posted: June 4, 2008
MathSciNet review: 2425750
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Abstract: We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann $\rho$ -invariants associated with certain metabelian representations, then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group.

References:

1.: A. J. Casson and C. McA. Gordon, Cobordism of classical knots, A la Recherche de la Topologie Perdue. Progress in Math., vol. 62 (Birkhäuser, 1986), pp. 181-199. MR 900252
2.: J. C. Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007), no. 885, x+95 pp. MR 2343079
3.: J. Cheeger and M. Gromov, Bounds on the von Neumann dimension of $L\sp 2$ -cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1-34. MR 806699 (87d:58136)
4.: T. D. Cochran, K. E. Orr and P. Teichner, Knot concordance, Whitney towers and -signatures, Ann. of Math. (2) 157 (2003), no. 2, 433-519. MR 1973052 (2004i:57003)
5.: T. D. Cochran, K. E. Orr and P. Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004), no. 1, 105-123. MR 2031301 (2004k:57005)
6.: S. Friedl, Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants, Algebr. Geom. Topol. 4 (2004), 893-934. MR 2100685 (2005j:57016)
7.: P. M. Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993), 1-19. MR 1201199 (94c:57007)
8.: R. E. Gompf and A. I. Stipsicz, -manifolds and Kirby calculus, Graduate Studies in Mathematics 20. American Mathematical Society, Providence, RI, 1999. MR 1707327 (2000h:57038)
9.: B. Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981), 189-192. MR 620010 (82h:57008)
10.: S.-G. Kim, Polynomial splittings of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 59-78. MR 2127228 (2005m:57018)
11.: T. Kim, Filtration of the classical knot concordance group and Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 293-306. MR 2092061 (2005f:57014)
12.: C. F. Letsche, An obstruction to slicing knots using the eta invariant, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 301-319. MR 1735303 (2001b:57017)
13.: J. Levine, Invariants of knot cobordism, Invent. Math. 8 (1969), 98-110. MR 0253348 (40:6563)
14.: J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229-244. MR 0246314 (39:7618)
15.: C. Livingston, Seifert forms and concordance, Geom. Topol. 6 (2002), 403-408. MR 1928840 (2003f:57019)
16.: D. Rolfsen, Knots and links (second printing), Mathematics Lecture Series, vol. 7 (Publish or Perish, Inc., Houston, Texas, 1976, 1990). MR 1277811 (95c:57018)
17.: C. T. C. Wall, Surgery on compact manifolds, Second edition. Edited and with a foreword by A. A. Ranicki. Mathematical Surveys and Monographs, 69. American Mathematical Society, Providence, RI, 1999. MR 1687388 (2000a:57089)

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