## Osaka Journal of Mathematics

### Rational Witt classes of pretzel knots

Stanislav Jabuka

#### Abstract

In his two pioneering articles [9, 10] Jerry Levine introduced and completely determined the algebraic concordance groups of odd dimensional knots. He did so by defining a host of invariants of algebraic concordance which he showed were a complete set of invariants. While being very powerful, these invariants are in practice often hard to determine, especially for knots with Alexander polynomials of high degree. We thus propose the study of a weaker set of invariants of algebraic concordance---the rational Witt classes of knots. Though these are rather weaker invariants than those defined by Levine, they have the advantage of lending themselves to quite manageable computability. We illustrate this point by computing the rational Witt classes of all pretzel knots. We give many examples and provide applications to obstructing sliceness for pretzel knots. Also, we obtain explicit formulae for the determinants and signatures of all pretzel knots. This article is dedicated to Jerry Levine and his lasting mathematical legacy; on the occasion of the conference Fifty years since Milnor and Fox'' held at Brandeis University on June 2--5, 2008.

#### Article information

Source
Osaka J. Math., Volume 47, Number 4 (2010), 977-1027.

Dates
First available in Project Euclid: 20 December 2010

https://projecteuclid.org/euclid.ojm/1292854315

Mathematical Reviews number (MathSciNet)
MR2791566

Zentralblatt MATH identifier
1210.57007

#### Citation

Jabuka, Stanislav. Rational Witt classes of pretzel knots. Osaka J. Math. 47 (2010), no. 4, 977--1027. https://projecteuclid.org/euclid.ojm/1292854315

#### References

• J.C. Cha: The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007).
• J.C. Cha and C. Livingston: KnotInfo: table of knot invariants, http://www.indiana.edu/ knotinfo, April 11, 2009.
• J. Greene and S. Jabuka: The slice-ribbon conjecture for 3-standed pretzel knots, preprint (2007), arXiv:0706.3398v2.
• J. Hillman: Algebraic Invariants of Links, Series on Knots and Everything 32, World Sci. Publishing, River Edge, NJ, 2002.
• J. Milnor and D. Husemoller: Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer, New Yor, 1973.
• M.A. Kervaire: Les nœ uds de dimensions supérieures, Bull. Soc. Math. France 93 (1965), 225--271.
• S.-G. Kim and C. Livingston: Knot mutation: 4-genus of knots and algebraic concordance, Pacific J. Math. 220 (2005), 87--105.
• T.Y. Lam: Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, Amer. Math. Soc., Providence, RI, 2005.
• J. Levine: Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229--244.
• J. Levine: Invariants of knot cobordism, Invent. Math. 8 (1969), 98--110.
• C. Livingston: A survey of classical knot concordance; in Handbook of Knot Theory, Elsevier B.V., Amsterdam, 319--347, 2005.
• C. Livingston: The algebraic concordance order of a knot, preprint (2008), arXiv:0806.3068.
• C. Livingston and S. Naik: Obstructing four-torsion in the classical knot concordance group, J. Differential Geom. 51 (1999), 1--12.
• T. Morita: Orders of knots in the algebraic knot cobordism group, Osaka J. Math. 25 (1988), 859--864.
• A. Pfister: Zur Darstellung von $-1$ als Summe von Quadraten in einem Körper, J. London Math. Soc. 40 (1965), 159--165.
• A. Pfister: Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), 116--132.
• D. Rolfsen: Knots and Links, Mathematics Lecture Series 7, Publish or Perish, Berkely, CA, 1976.
• O.T. O'Meara: Introduction to Quadratic Forms, Classics in Mathematics, Springer, Berlin, 2000.
• A. Stoimenow: Some examples related to knot sliceness, J. Pure Appl. Algebra 210 (2007), 161--175.
• E. Witt: Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31--44.