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  1. ; ; ; ; ; (, Michigan Mathematical Journal)
    This paper investigates the exotic phenomena exhibited by links of disconnected surfaces with boundary that are properly embedded in the 4-ball. Our main results provide two different constructions of exotic pairs of surface links that are Brunnian, meaning that all proper sublinks of the surface are trivial. We then modify these core constructions to vary the number of components in the exotic links, the genera of the components, and the number of components that must be removed before the surfaces become unlinked. Our arguments extend two tools from 3-dimensional knot theory into the 4-dimensional setting: satellite operations, especially Bing doubling, and covering links in branched covers. 
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  2. ; (, Pacific journal of mathematics)
    We show that for every odd prime $$q$$, there exists an infinite family~$$\{M_i\}_{i=1}^{\infty}$$ of topological 4-manifolds that are all stably homeomorphic to one another, all the manifolds $$M_i$$ have isometric rank one equivariant intersection pairings and boundary $$L(2q, 1) \# (S^1 \times S^2)$$, but they are pairwise not homotopy equivalent via any homotopy equivalence that restricts to a homotopy equivalence of the boundary. 
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    Accepted Manuscript Full Text Available
  3. ; ; ; ; ; (, Compositio Mathematica)
    Abstract The trace of the $$n$$ -framed surgery on a knot in $$S^{3}$$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $$2$$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $$3$$ -dimensional knot invariants. For each $$n$$ , this provides conditions that imply a knot is topologically $$n$$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice. 
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  4. ; (, L’Enseignement Mathématique)
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