Knot concordance and homology cobordism

Cochran, Tim; Franklin, Bridget; Hedden, Matthew; Horn, Peter

doi:10.1090/S0002-9939-2013-11471-1

Knot concordance and homology cobordism
HTML articles powered by AMS MathViewer

by Tim D. Cochran, Bridget D. Franklin, Matthew Hedden and Peter D. Horn PDF

Proc. Amer. Math. Soc. 141 (2013), 2193-2208 Request permission

Abstract:

We consider the question: “If the zero-framed surgeries on two oriented knots in $S^3$ are $\mathbb {Z}$-homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?” We show that this question has a negative answer in the smooth category, even for topologically slice knots. To show this we first prove that the zero-framed surgery on $K$ is $\mathbb {Z}$-homology cobordant to the zero-framed surgery on many of its winding number one satellites $P(K)$. Then we prove that in many cases the $\tau$ and $s$-invariants of $K$ and $P(K)$ differ. Consequently neither $\tau$ nor $s$ is an invariant of the smooth homology cobordism class of the zero-framed surgery. We also show that a natural rational version of this question has a negative answer in both the topological and smooth categories by proving similar results for $K$ and its $(p,1)$-cables.

References

Selman Akbulut and Rostislav Matveyev, Exotic structures and adjunction inequality, Turkish J. Math. 21 (1997), no. 1, 47–53. MR 1456158
Jae Choon Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007), no. 885, x+95. MR 2343079, DOI 10.1090/memo/0885
Jae Choon Cha and Ki Hyoung Ko, Signatures of links in rational homology spheres, Topology 41 (2002), no. 6, 1161–1182. MR 1923217, DOI 10.1016/S0040-9383(01)00029-5
Jae Choon Cha and Ki Hyoung Ko, Signature invariants of covering links, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3399–3412. MR 2218981, DOI 10.1090/S0002-9947-05-03739-6
Jae Choon Cha, Charles Livingston, and Daniel Ruberman, Algebraic and Heegaard-Floer invariants of knots with slice Bing doubles, Math. Proc. Cambridge Philos. Soc. 144 (2008), no. 2, 403–410. MR 2405897, DOI 10.1017/S0305004107000795
Tim D. Cochran, Geometric invariants of link cobordism, Comment. Math. Helv. 60 (1985), no. 2, 291–311. MR 800009, DOI 10.1007/BF02567416
Tim D. Cochran, Shelly Harvey, and Constance Leidy, Knot concordance and higher-order Blanchfield duality, Geom. Topol. 13 (2009), no. 3, 1419–1482. MR 2496049, DOI 10.2140/gt.2009.13.1419
Tim D. Cochran, Shelly Harvey, and Constance Leidy, 2-torsion in the $n$-solvable filtration of the knot concordance group, Proc. Lond. Math. Soc. (3) 102 (2011), no. 2, 257–290. MR 2769115, DOI 10.1112/plms/pdq020
T. D. Cochran and K. E. Orr, Not all links are concordant to boundary links, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 99–106. MR 1031581, DOI 10.1090/S0273-0979-1990-15912-9
Tim D. Cochran and Kent E. Orr, Not all links are concordant to boundary links, Ann. of Math. (2) 138 (1993), no. 3, 519–554. MR 1247992, DOI 10.2307/2946555
Tim D. Cochran, Kent E. Orr, and Peter Teichner, Knot concordance, Whitney towers and $L^2$-signatures, Ann. of Math. (2) 157 (2003), no. 2, 433–519. MR 1973052, DOI 10.4007/annals.2003.157.433
John B. Etnyre, Legendrian and transversal knots, Handbook of knot theory, Elsevier B. V., Amsterdam, 2005, pp. 105–185. MR 2179261, DOI 10.1016/B978-044451452-3/50004-6
Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), no. 3, 357–453. MR 679066
Michael Freedman, Robert Gompf, Scott Morrison, and Kevin Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topol. 1 (2010), no. 2, 171–208. MR 2657647, DOI 10.4171/QT/5
Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
Patrick M. Gilmer, Link cobordism in rational homology $3$-spheres, J. Knot Theory Ramifications 2 (1993), no. 3, 285–320. MR 1238876, DOI 10.1142/S0218216593000179
Robert E. Gompf and András I. Stipsicz, $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR 1707327, DOI 10.1090/gsm/020
Matthew Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338. MR 2372849, DOI 10.2140/gt.2007.11.2277
Matthew Hedden, On knot Floer homology and cabling. II, Int. Math. Res. Not. IMRN 12 (2009), 2248–2274. MR 2511910, DOI 10.1093/imrn/rnp015
Jonathan A. Hillman, Alexander ideals of links, Lecture Notes in Mathematics, vol. 895, Springer-Verlag, Berlin-New York, 1981. MR 653808, DOI 10.1007/BFb0091682
Akio Kawauchi, On links not cobordant to split links, Topology 19 (1980), no. 4, 321–334. MR 584558, DOI 10.1016/0040-9383(80)90017-8
Akio Kawauchi, Rational-slice knots via strongly negative-amphicheiral knots, Commun. Math. Res. 25 (2009), no. 2, 177–192. MR 2554510
C. Kearton, The Milnor signatures of compound knots, Proc. Amer. Math. Soc. 76 (1979), no. 1, 157–160. MR 534409, DOI 10.1090/S0002-9939-1979-0534409-1
Rob Kirby (ed.), Problems in low-dimensional topology, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 35–473. MR 1470751, DOI 10.1090/amsip/002.2/02
J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
W. B. Raymond Lickorish, An introduction to knot theory, Graduate Texts in Mathematics, vol. 175, Springer-Verlag, New York, 1997. MR 1472978, DOI 10.1007/978-1-4612-0691-0
R. A. Litherland, Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977) Lecture Notes in Math., vol. 722, Springer, Berlin, 1979, pp. 71–84. MR 547456
R. A. Litherland, Cobordism of satellite knots, Four-manifold theory (Durham, N.H., 1982) Contemp. Math., vol. 35, Amer. Math. Soc., Providence, RI, 1984, pp. 327–362. MR 780587, DOI 10.1090/conm/035/780587
Charles Livingston, Knots which are not concordant to their reverses, Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135, 323–328. MR 711524, DOI 10.1093/qmath/34.3.323
Lenhard Ng, The Legendrian satellite construction, preprint, http://arxiv.org/abs/math/ 0112105.
Lenhard Ng, Peter Ozsváth, and Dylan Thurston, Transverse knots distinguished by knot Floer homology, J. Symplectic Geom. 6 (2008), no. 4, 461–490. MR 2471100, DOI 10.4310/JSG.2008.v6.n4.a4
Lenhard Ng and Lisa Traynor, Legendrian solid-torus links, J. Symplectic Geom. 2 (2004), no. 3, 411–443. MR 2131643, DOI 10.4310/JSG.2004.v2.n3.a6
Burak Ozbagci and András I. Stipsicz, Surgery on contact 3-manifolds and Stein surfaces, Bolyai Society Mathematical Studies, vol. 13, Springer-Verlag, Berlin; János Bolyai Mathematical Society, Budapest, 2004. MR 2114165, DOI 10.1007/978-3-662-10167-4
Peter Ozsváth and Zoltán Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003), 615–639. MR 2026543, DOI 10.2140/gt.2003.7.615
Olga Plamenevskaya, Bounds for the Thurston-Bennequin number from Floer homology, Algebr. Geom. Topol. 4 (2004), 399–406. MR 2077671, DOI 10.2140/agt.2004.4.399
Olga Plamenevskaya, Transverse knots and Khovanov homology, Math. Res. Lett. 13 (2006), no. 4, 571–586. MR 2250492, DOI 10.4310/MRL.2006.v13.n4.a7
Jacob Andrew Rasmussen, Floer homology and knot complements, ProQuest LLC, Ann Arbor, MI, 2003. Thesis (Ph.D.)–Harvard University. MR 2704683
Jacob Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010), no. 2, 419–447. MR 2729272, DOI 10.1007/s00222-010-0275-6
Lee Rudolph, An obstruction to sliceness via contact geometry and “classical” gauge theory, Invent. Math. 119 (1995), no. 1, 155–163. MR 1309974, DOI 10.1007/BF01245177
Lee Rudolph, The slice genus and the Thurston-Bennequin invariant of a knot, Proc. Amer. Math. Soc. 125 (1997), no. 10, 3049–3050. MR 1443854, DOI 10.1090/S0002-9939-97-04258-5
Alexander N. Shumakovitch, Rasmussen invariant, slice-Bennequin inequality, and sliceness of knots, J. Knot Theory Ramifications 16 (2007), no. 10, 1403–1412. MR 2384833, DOI 10.1142/S0218216507005889