Celebratio Mathematica

A. J. Casson and C. M. Gordon: “On slice knots in dimension three,” pp. 39–53 in Algebraic and geometric topology (Stanford, CA, 2–21 August 1976), Part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics 32. American Mathematical Society (Providence, RI), 1978. MR 520521 Zbl 0394.57008 incollection

Under the equivalence relation of concordance (sometimes called cobordism), smooth knots in the 3-sphere \( S^3 \) form an abelian group with respect to connected sum [Fox and Milnor 1966]. The knots \( K \) representing the zero class are precisely those which are slice, that is, satisfy \[ (S^3,K) = \partial(B^4,D) \] for some smooth 2-disc \( D \) in the 4-ball \( B^4 \), Now associated with any knot \( K \) and a Seifert surface \( V \) spanning \( K \), is a bilinear Seifert pairing \[ \theta_V: H_1(V)\times H_1(V)\to \mathbf{Z} \] [Seifert 1935; Levine 1969]. We say that \( K \) is algebraically slice if \( \theta_V \) vanishes on a subgroup of \( H_1(V) \) whose rank is \( \frac{1}{2} \)rank \( H_1(V) \) (this condition is independent of the choice of \( V \)). It is known that a necessary condition for \( K \) to be slice is that it be algebraically slice. Moreover, in higher (odd) dimensions analogous definitions may be made, and there the conditions are equivalent [Levine 1969]. We shall show that this is not the case in dimension 3.

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A. J. Casson and C. M. Gordon: “A loop theorem for duality spaces and fibred ribbon knots,” Invent. Math. 74 : 1 (February 1983), pp. 119–137. MR 722728 Zbl 0538.57003 article

A. J. Casson and C. M. Gordon: “Cobordism of classical knots,” pp. 181–199 in À la recherche de la topologie perdue [In search of the lost topology]. Edited by L. Guillou and A. Marin. Progress in Mathematics 62. Birkhäuser (Boston), 1986. With an appendix by P. M. Gilmer. MR 900252 incollection

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A. J. Casson and C. M. Gordon: “Reducing Heegaard splittings,” Topology Appl. 27 : 3 (December 1987), pp. 275–283. MR 918537 Zbl 0632.57010 article

Proceedings of the Casson Fest (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003), published as Geom. Topol. Monogr. 7. Issue edited by C. Gordon and Y. Rieck. Geometry & Topology Publications (Coventry, UK), 2004. MR 2169266 Zbl 1066.57002 book

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Cameron McAllan Gordon

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