Mathematical Research Letters

Volume 13 (2006)

Number 2

A characterisation of the $\mathbf{n\langle1\rangle \oplus\langle3\rangle}$ form and applications to rational homology spheres

Pages: 259 – 271



Brendan Owens (Louisiana State University)

Saso Strle (University of Ljubljana)


We conjecture two generalisations of Elkies' theorem on unimodular quadratic forms to non-unimodular forms. We give some evidence for these conjectures including a result for determinant 3. These conjectures, when combined with results of \froyshov~ and of \ozsvath~ and \szabo, would give a simple test of whether a rational homology 3-sphere may bound a negative-definite four-manifold. We verify some predictions using Donaldson’s theorem. Based on this we compute the four-ball genus of some Montesinos knots.

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