Linear independence of cables in the knot concordance group
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- by Christopher W. Davis, JungHwan Park and Arunima Ray PDF
- Trans. Amer. Math. Soc. 374 (2021), 4449-4479 Request permission
Abstract:
We produce infinite families of knots $\{K^i\}_{i\ge 1}$ for which the set of cables $\{K^i_{p,1}\}_{i,p\ge 1}$ is linearly independent in the knot concordance group, $\mathcal {C}$. We arrange that these examples lie arbitrarily deep in the solvable and bipolar filtrations of $\mathcal {C}$, denoted by $\{\mathcal {F}_n\}$ and $\{\mathcal {B}_n\}$ respectively. As a consequence, this result cannot be reached by any combination of algebraic concordance invariants, Casson-Gordon invariants, and Heegaard-Floer invariants such as $\tau$, $\varepsilon$, and $\Upsilon$. We give two applications of this result. First, for any $n\ge 0$, there exists an infinite family $\{K^i\}_{i\geq 1}$ such that for each fixed $i$, $\{K^i_{2^j,1}\}_{j\geq 0}$ is a basis for an infinite rank summand of $\mathcal {F}_n$ and $\{K^i_{p,1}\}_{i, p\geq 1}$ is linearly independent in $\mathcal {F}_{n}/\mathcal {F}_{n.5}$. Second, for any $n\ge 1$, we give filtered counterexamples to Kauffman’s conjecture on slice knots by constructing smoothly slice knots with genus one Seifert surfaces where one derivative curve has nontrivial Arf invariant and the other is nontrivial in both $\mathcal {F}_n/\mathcal {F}_{n.5}$ and $\mathcal {B}_{n-1}/\mathcal {B}_{n+1}$. We also give examples of smoothly slice knots with genus one Seifert surfaces such that one derivative has nontrivial Arf invariant and the other is topologically slice but not smoothly slice.References
- Evan M. Bullock and Christopher William Davis, Strong coprimality and strong irreducibility of Alexander polynomials, Topology Appl. 159 (2012), no. 1, 133–143. MR 2852954, DOI 10.1016/j.topol.2011.08.019
- Nicolae Ciprian Bonciocat, Schönemann-Eisenstein-Dumas-type irreducibility conditions that use arbitrarily many prime numbers, Comm. Algebra 43 (2015), no. 8, 3102–3122. MR 3354081, DOI 10.1080/00927872.2014.910800
- Tim D. Cochran and Christopher William Davis, Counterexamples to Kauffman’s conjectures on slice knots, Adv. Math. 274 (2015), 263–284. MR 3318151, DOI 10.1016/j.aim.2014.12.006
- 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 10.1090/S0002-9947-07-04176-1
- Tim D. Cochran, Bridget D. Franklin, Matthew Hedden, and Peter D. Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013), no. 6, 2193–2208. MR 3034445, DOI 10.1090/S0002-9939-2013-11471-1
- Jeff Cheeger and Mikhael Gromov, Bounds on the von Neumann dimension of $L^2$-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), no. 1, 1–34. MR 806699
- Tim D. Cochran and Peter D. Horn, Structure in the bipolar filtration of topologically slice knots, Algebr. Geom. Topol. 15 (2015), no. 1, 415–428. MR 3325742, DOI 10.2140/agt.2015.15.415
- Jae Choon Cha, Topological minimal genus and $L^2$-signatures, Algebr. Geom. Topol. 8 (2008), no. 2, 885–909. MR 2443100, DOI 10.2140/agt.2008.8.885
- W. Chen, On the Upsilon invariant of cable knots, 2016. Preprint: http://arxiv.org/abs/1604.04760.
- Tim D. Cochran, Shelly Harvey, and Peter Horn, Filtering smooth concordance classes of topologically slice knots, Geom. Topol. 17 (2013), no. 4, 2103–2162. MR 3109864, DOI 10.2140/gt.2013.17.2103
- Tim Cochran, Shelly Harvey, and Constance Leidy, Link concordance and generalized doubling operators, Algebr. Geom. Topol. 8 (2008), no. 3, 1593–1646. MR 2443256, DOI 10.2140/agt.2008.8.1593
- 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
- Tim D. Cochran, Shelly Harvey, and Constance Leidy, Primary decomposition and the fractal nature of knot concordance, Math. Ann. 351 (2011), no. 2, 443–508. MR 2836668, DOI 10.1007/s00208-010-0604-5
- J. C. Cha and M. H. Kim, The bipolar filtration of topologically slice knots, 2017. Preprint: http://arxiv.org/abs/1710.07803.
- Tim D. Cochran, Noncommutative knot theory, Algebr. Geom. Topol. 4 (2004), 347–398. MR 2077670, DOI 10.2140/agt.2004.4.347
- 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
- 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, DOI 10.1007/s00014-001-0793-6
- Jae Choon Cha and Mark Powell, Covering link calculus and the bipolar filtration of topologically slice links, Geom. Topol. 18 (2014), no. 3, 1539–1579. MR 3228458, DOI 10.2140/gt.2014.18.1539
- Tim D. Cochran and Peter Teichner, Knot concordance and von Neumann $\rho$-invariants, Duke Math. J. 137 (2007), no. 2, 337–379. MR 2309149, DOI 10.1215/S0012-7094-07-13723-2
- Christopher William Davis, First Order Signatures and Knot Concordance, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)–Rice University. MR 3130814
- Christopher William Davis, Von Neumann rho invariants and torsion in the topological knot concordance group, Algebr. Geom. Topol. 12 (2012), no. 2, 753–789. MR 2914617, DOI 10.2140/agt.2012.12.753
- David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
- Irving Dai, Jennifer Hom, Matthew Stoffregen, and Linh Truong, More concordance homomorphisms from knot floer homology, 2019.
- C. W. Davis, J. Park, and A. Ray, Handlebody solvable knots, 2020+. In preparation.
- 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
- Ralph H. Fox and John W. Milnor, Singularities of $2$-spheres in $4$-space and cobordism of knots, Osaka Math. J. 3 (1966), 257–267. MR 211392
- Peter Feller, JungHwan Park, and Arunima Ray, On the Upsilon invariant and satellite knots, Math. Z. 292 (2019), no. 3-4, 1431–1452. MR 3980298, DOI 10.1007/s00209-018-2145-7
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- Michael H. Freedman, A surgery sequence in dimension four; the relations with knot concordance, Invent. Math. 68 (1982), no. 2, 195–226. MR 666159, DOI 10.1007/BF01394055
- Stavros Garoufalidis and Peter Teichner, On knots with trivial Alexander polynomial, J. Differential Geom. 67 (2004), no. 1, 167–193. MR 2153483
- 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
- Jennifer Hom, Bordered Heegaard Floer homology and the tau-invariant of cable knots, J. Topol. 7 (2014), no. 2, 287–326. MR 3217622, DOI 10.1112/jtopol/jtt030
- Jennifer Hom, The knot Floer complex and the smooth concordance group, Comment. Math. Helv. 89 (2014), no. 3, 537–570. MR 3260841, DOI 10.4171/CMH/326
- Jennifer Hom, An infinite-rank summand of topologically slice knots, Geom. Topol. 19 (2015), no. 2, 1063–1110. MR 3336278, DOI 10.2140/gt.2015.19.1063
- Jennifer Hom, Correction to the article An infinite-rank summand of topologically slice knots, Geom. Topol. 23 (2019), no. 5, 2699–2700. MR 4019902, DOI 10.2140/gt.2019.23.2699
- Jennifer Hom and Zhongtao Wu, Four-ball genus bounds and a refinement of the Ozváth-Szabó tau invariant, J. Symplectic Geom. 14 (2016), no. 1, 305–323. MR 3523259, DOI 10.4310/JSG.2016.v14.n1.a12
- Bo Ju Jiang, A simple proof that the concordance group of algebraically slice knots is infinitely generated, Proc. Amer. Math. Soc. 83 (1981), no. 1, 189–192. MR 620010, DOI 10.1090/S0002-9939-1981-0620010-7
- Louis H. Kauffman, On knots, Annals of Mathematics Studies, vol. 115, Princeton University Press, Princeton, NJ, 1987. MR 907872
- 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
- C. Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. (2) 10 (1975), no. 4, 406–408. MR 385873, DOI 10.1112/jlms/s2-10.4.406
- Se-Goo Kim, Polynomial splittings of Casson-Gordon invariants, Math. Proc. Cambridge Philos. Soc. 138 (2005), no. 1, 59–78. MR 2127228, DOI 10.1017/S0305004104008023
- Min Hoon Kim and Kyungbae Park, An infinite-rank summand of knots with trivial Alexander polynomial, J. Symplectic Geom. 16 (2018), no. 6, 1749–1771. MR 3934241, DOI 10.4310/JSG.2018.v16.n6.a5
- Wolfgang Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649, DOI 10.1007/978-3-662-04687-6
- J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244. MR 246314, DOI 10.1007/BF02564525
- 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, Computations of the Ozsváth-Szabó knot concordance invariant, Geom. Topol. 8 (2004), 735–742. MR 2057779, DOI 10.2140/gt.2004.8.735
- Charles Livingston, Order 2 algebraically slice knots, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 335–342. MR 1734416, DOI 10.2140/gtm.1999.2.335
- Charles Livingston and Paul Melvin, Abelian invariants of satellite knots, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 217–227. MR 827271, DOI 10.1007/BFb0075225
- Wolfgang Lück and Thomas Schick, Various $L^2$-signatures and a topological $L^2$-signature theorem, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 362–399. MR 2048728, DOI 10.1142/9789812704443_{0}015
- Wolfgang Lück, $L^2$-invariants: theory and applications to geometry and $K$-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 44, Springer-Verlag, Berlin, 2002. MR 1926649, DOI 10.1007/978-3-662-04687-6
- Preda Mihăilescu, Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math. 572 (2004), 167–195. MR 2076124, DOI 10.1515/crll.2004.048
- L. Ng, The Legendrian satellite construction, 2001. Preprint: http://arxiv.org/abs/0112105.
- Lenhard Ng and Lisa Traynor, Legendrian solid-torus links, J. Symplectic Geom. 2 (2004), no. 3, 411–443. MR 2131643
- Yi Ni and Zhongtao Wu, Cosmetic surgeries on knots in $S^3$, J. Reine Angew. Math. 706 (2015), 1–17. MR 3393360, DOI 10.1515/crelle-2013-0067
- 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
- Peter Ozsváth and Zoltán Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027–1158. MR 2113019, DOI 10.4007/annals.2004.159.1027
- Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Concordance homomorphisms from knot Floer homology, Adv. Math. 315 (2017), 366–426. MR 3667589, DOI 10.1016/j.aim.2017.05.017
- JungHwan Park, A construction of slice knots via annulus modifications, Topology Appl. 238 (2018), 1–19. MR 3775112, DOI 10.1016/j.topol.2018.01.010
- 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
- JungHwan Park and Arunima Ray, A family of non-split topologically slice links with arbitrarily large smooth slice genus, Proc. Amer. Math. Soc. 146 (2018), no. 1, 439–448. MR 3723153, DOI 10.1090/proc/13724
- Arunima Ray, Satellite operators with distinct iterates in smooth concordance, Proc. Amer. Math. Soc. 143 (2015), no. 11, 5005–5020. MR 3391056, DOI 10.1090/proc/12625
- H. Seifert, On the homology invariants of knots, Quart. J. Math. Oxford Ser. (2) 1 (1950), 23–32. MR 35436, DOI 10.1093/qmath/1.1.23
Additional Information
- Christopher W. Davis
- Affiliation: Department of Mathematics, University of Wisconsin–Eau Claire, 105 Garfield Avenue P.O. Box 4004, Eau Claire, Wisconsin 54702
- MR Author ID: 958152
- Email: daviscw@uwec.edu
- JungHwan Park
- Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, South Korea
- MR Author ID: 1188099
- Email: jungpark0817@gmail.com
- Arunima Ray
- Affiliation: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 1039665
- Email: aruray@gmail.com
- Received by editor(s): May 6, 2019
- Received by editor(s) in revised form: November 10, 2020
- Published electronically: February 11, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4449-4479
- MSC (2020): Primary 57K10
- DOI: https://doi.org/10.1090/tran/8336
- MathSciNet review: 4251235