Celebratio Mathematica — Gordon

Cameron Gordon: a brief biography

Works connected to Richard James Milgram

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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.

Andrew J. Casson	Related
Richard James Milgram	Related