Available in electronic format
Available in print format		 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article
     

On slicing invariants of knots

Author(s): Brendan Owens
Journal: Trans. Amer. Math. Soc.
MSC (2000): Primary 57M25
Posted: August 13, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: The slicing number of a knot, $ u_s(K)$, is the minimum number of crossing changes required to convert $ K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $ g_s(K)$. We show that for many knots, previous bounds on the unknotting number obtained by Ozsváth and Szabó and by the author in fact give bounds on the slicing number. Livingston defined another invariant $ U_s(K)$, which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus. We exhibit an infinite family of knots $ K_n$ with slice genus $ n$ and Livingston invariant greater than $ n$. Our bounds are based on restrictions (using Donaldson's diagonalisation theorem or Heegaard Floer homology) on the intersection forms of four-manifolds bounded by the double branched cover of a knot.

References:

1.
J. C. Cha & C. Livingston, Table of knot invariants, http://www.indiana.edu/~knotinfo.

2.
A. J. Casson & C. McA. Gordon, On slice knots in dimension three, Algebraic and geometric topology, Part 2, Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc., 1978, 39-53. MR 520521 (81g:57003)

3.
T. D. Cochran & W. B. R. Lickorish, Unknotting information from $ 4$-manifolds, Trans. Amer. Math. Soc. 297 1986, 125-142. MR 849471 (87i:57003)

4.
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Diff. Geom. 18 1983, 279-315. MR 710056 (85c:57015)

5.
C. McA. Gordon & R. A. Litherland, On the signature of a link, Invent. Math. 47 1978, 53-69. MR 0500905 (58:18407)

6.
R. E. Gompf & A. I. Stipsicz, $ 4$-manifolds and Kirby calculus, Graduate Studies in Math. 20, Amer. Math. Soc., 1999. MR 1707327 (2000h:57038)

7.
P. M. Gruber & C. G. Lekkerkerker, Geometry of numbers, 2nd ed., North-Holland Mathematical Library 37, North-Holland, 1987. MR 893813 (88j:11034)

8.
W. B. R. Lickorish, The unknotting number of a classical knot, Contemp. Math. 44 1985, 117-121. MR 813107 (87a:57012)

9.
W. B. R. Lickorish, An introduction to knot theory, Graduate Texts in Math. 175, Springer, 1997. MR 1472978 (98f:57015)

10.
P. Lisca, Symplectic fillings and positive scalar curvature, Geometry and Topology 2 1998, 103-116. MR 1633282 (99f:57038)

11.
P. Lisca, Lens spaces, rational balls and the ribbon conjecture, Geometry and Topology 11 2007, 429-472. MR 2302495 (2008a:57008)

12.
C. Livingston, The slicing number of a knot, Algebraic and Geometric Topology 2 2002, 1051-1060. MR 1936979 (2003m:57018)

13.
J. M. Montesinos, Variedades de Seifert que son cubiertas ciclicas ramificados de dos hojas, Bol. Soc. Mat. Mexicana 18 1973, 1-32. MR 0341467 (49:6218)

14.
H. Murakami & A. Yasuhara, Four-genus and four-dimensional clasp number of a knot, Proc. Amer. Math. Soc. 128 2000, 3693-3699. MR 1690998 (2001b:57020)

15.
K. Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 1965, 387-422. MR 0171275 (30:1506)

16.
T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Invariants of knots and 3-manifolds (Kyoto 2001), Geometry and Topology Monographs 4 2002, 377-572. MR 2065029 (2005c:57014)

17.
B. Owens, Unknotting information from Heegaard Floer homology, Advances in Mathematics, 217 (2008), pages 2353-2376. MR 2388097

18.
B. Owens & S. Strle, Rational homology spheres and the four-ball genus of knots, Advances in Mathematics 200, 2006, 196-216. MR 2199633 (2006m:57016)

19.
B. Owens & S. Strle, A characterisation of the $ n\langle1\rangle\oplus\langle3\rangle$ form and applications to rational homology spheres, Math. Res. Lett. 13, 2006, 259-271. MR 2231116 (2007j:57036)

20.
P. Ozsváth & Z. Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Advances in Mathematics 173 2003, 179-261. MR 1957829 (2003m:57066)

21.
P. Ozsváth & Z. Szabó, On the Floer homology of plumbed three-manifolds, Geometry and Topology 7 2003, 225-254. MR 1988285 (2004f:57040)

22.
P. Ozsváth & Z. Szabó, Knots with unknotting number one and Heegaard Floer homology, Topology 44 2005, 705-745. MR 2136532 (2005m:57045)

23.
L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv. 58 1983, 1-37. MR 699004 (84j:57006)

24.
A. Stoimenow, Polynomial values, the linking form and unknotting numbers, Math. Res. Lett. 11 2004, 755-769. MR 2106240 (2005j:57022)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57M25

Retrieve articles in all Journals with MSC (2000): 57M25

Additional Information:

Brendan Owens
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Address at time of publication: Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, United Kingdom

DOI: 10.1090/S0002-9947-09-04904-6
PII: S 0002-9947(09)04904-6
Received by editor(s): April 11, 2008
Posted: August 13, 2009
Additional Notes: The author was supported in part by NSF grant DMS-0604876.
Copyright of article: Copyright 2009, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google