Satellites and concordance of knots in 3–manifolds
Authors:
Stefan Friedl, Matthias Nagel, Patrick Orson and Mark Powell
Journal:
Trans. Amer. Math. Soc. 371 (2019), 2279-2306
MSC (2010):
Primary 57M27, 57N70
DOI:
https://doi.org/10.1090/tran/7313
Published electronically:
September 10, 2018
MathSciNet review:
3896081
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Abstract | References | Similar Articles | Additional Information
Abstract: Given a $3$–manifold $Y$ and a free homotopy class in $[S^1,Y]$, we investigate the set of topological concordance classes of knots in $Y \times [0,1]$ representing the given homotopy class. The concordance group of knots in the $3$–sphere acts on this set. We show in many cases that the action is not transitive, using two techniques. Our first technique uses Reidemeister torsion invariants, and the second uses linking numbers in covering spaces. In particular, we show using covering links that for the trivial homotopy class, and for any $3$–manifold that is not the $3$–sphere, the set of orbits is infinite. On the other hand, for the case that $Y=S^1 \times S^2$, we apply topological surgery theory to show that all knots with winding number one are concordant.
- Matthias Aschenbrenner, Stefan Friedl, and Henry Wilton, 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015. MR 3444187
- Daniele Celoria, On concordances in 3-manifolds, J. Topol. 11 (2018), no. 1, 180–200. MR 3784229, DOI https://doi.org/10.1112/topo.12051
- David Cimasoni and Vincent Florens, Generalized Seifert surfaces and signatures of colored links, Trans. Amer. Math. Soc. 360 (2008), no. 3, 1223–1264. MR 2357695, DOI https://doi.org/10.1090/S0002-9947-07-04176-1
- Jae Choon Cha and Stefan Friedl, Twisted torsion invariants and link concordance, Forum Math. 25 (2013), no. 3, 471–504. MR 3062861, DOI https://doi.org/10.1515/form.2011.125
- T. A. Chapman, Topological invariance of Whitehead torsion, Amer. J. Math. 96 (1974), 488–497. MR 391109, DOI https://doi.org/10.2307/2373556
- D. Cooper, The universal abelian cover of a link, Low-dimensional topology (Bangor, 1979) London Math. Soc. Lecture Note Ser., vol. 48, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 51–66. MR 662427
- Jae Choon Cha and Mark Powell, Nonconcordant links with homology cobordant zero-framed surgery manifolds, Pacific J. Math. 272 (2014), no. 1, 1–33. MR 3270170, DOI https://doi.org/10.2140/pjm.2014.272.1
- Sylvain E. Cappell and Julius L. Shaneson, The codimension two placement problem and homology equivalent manifolds, Ann. of Math. (2) 99 (1974), 277–348. MR 339216, DOI https://doi.org/10.2307/1970901
- James F. Davis, A two component link with Alexander polynomial one is concordant to the Hopf link, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 2, 265–268. MR 2212279, DOI https://doi.org/10.1017/S0305004105008856
- Stefan Friedl and Taehee Kim, Twisted Alexander norms give lower bounds on the Thurston norm, Trans. Amer. Math. Soc. 360 (2008), no. 9, 4597–4618. MR 2403698, DOI https://doi.org/10.1090/S0002-9947-08-04455-3
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Stefan Friedl and Stefano Vidussi, A survey of twisted Alexander polynomials, The mathematics of knots, Contrib. Math. Comput. Sci., vol. 1, Springer, Heidelberg, 2011, pp. 45–94. MR 2777847, DOI https://doi.org/10.1007/978-3-642-15637-3_3
- Deborah L. Goldsmith, A linking invariant of classical link concordance, Knot theory (Proc. Sem., Plans-sur-Bex, 1977) Lecture Notes in Math., vol. 685, Springer, Berlin, 1978, pp. 135–170. MR 521732
- Prudence Heck, Homotopy properties of knots in prime manifolds, preprint, available at https://arxiv.org/abs/1110.6903, 2011.
- Jonathan Hillman, Algebraic invariants of links, 2nd ed., Series on Knots and Everything, vol. 52, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. MR 2931688
- W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978
- John Milnor, Isotopy of links. Algebraic geometry and topology, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N. J., 1957, pp. 280–306. MR 0092150
- David Miller, An extension of Milnor’s $\overline \mu $-invariants, Topology Appl. 65 (1995), no. 1, 69–82. MR 1354382, DOI https://doi.org/10.1016/0166-8641%2894%2900098-N
- Dale Rolfsen, Piecewise-linear $I$-equivalence of links, Low-dimensional topology (Chelwood Gate, 1982) London Math. Soc. Lecture Note Ser., vol. 95, Cambridge Univ. Press, Cambridge, 1985, pp. 161–178. MR 827301, DOI https://doi.org/10.1017/CBO9780511662744.006
- Rob Schneiderman, Algebraic linking numbers of knots in 3-manifolds, Algebr. Geom. Topol. 3 (2003), 921–968. MR 2012959, DOI https://doi.org/10.2140/agt.2003.3.921
- Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
- John Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170–181. MR 175956, DOI https://doi.org/10.1016/0021-8693%2865%2990017-7
- Vladimir Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001. Notes taken by Felix Schlenk. MR 1809561
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Additional Information
Stefan Friedl
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
MR Author ID:
746949
Email:
stefan.friedl@mathematik.uni-regensburg.de
Matthias Nagel
Affiliation:
Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
Email:
nagel@cirget.ca
Patrick Orson
Affiliation:
Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
MR Author ID:
1157713
Email:
patrick.orson@cirget.ca
Mark Powell
Affiliation:
Département de Mathématiques, Université du Québec, Montréal, Québec, H2X 3YZ Canada
MR Author ID:
975189
Email:
mark@cirget.ca
Received by editor(s):
December 6, 2016
Received by editor(s) in revised form:
June 19, 2017
Published electronically:
September 10, 2018
Additional Notes:
The first author was supported by the SFB 1085 ‘Higher Invariants’ at the University of Regensburg, funded by the Deutsche Forschungsgemeinschaft (DFG). The first author is grateful for the hospitality received at Durham University and wishes to thank Wolfgang Lück for supporting a long stay at the Hausdorff Institute.
The second author was supported by a CIRGET postdoctoral fellowship.
The third author was supported by the EPSRC grant EP/M000389/1 of Andrew Lobb.
The fourth author was supported by an NSERC Discovery Grant.
The second, third, and fourth authors all thank the Hausdorff Institute for Mathematics in Bonn for both support and its outstanding research environment.
Article copyright:
© Copyright 2018
American Mathematical Society
