Celebratio Mathematica

C. M. Gordon and R. A. Litherland: “On the signature of a link,” Invent. Math. 47 : 1 (February 1978), pp. 53–69. MR 500905 Zbl 0391.57004 article

C. M. Gordon and R. A. Litherland: “On a theorem of Murasugi,” Pac. J. Math. 82 : 1 (1979), pp. 69–74. MR 549833 Zbl 0391.57003 article

Let \( l = k_1\cup k_2 \) be a 2-component link in \( S^3 \), with \( k_2 \) unknotted. The 2-fold cover of \( S^3 \) branched over \( k_2 \) is again \( S^3 \); let \( k_1^{(2)} \) be the inverse image of \( k_1 \), and suppose that \( k_1^{(2)} \) is connected. How are the signatures \( \sigma(k_1) \), \( \sigma(k_1^{(2)}) \) of the knots \( k_1 \) and \( k_1^{(2)} \) related? This question was considered (from a slightly different point of view) by Murasugi, who gave the following answer [Topology, 9 (1970), 283–298].

Theorem 1 (Murasugi). \[ \sigma(k_1^{(2)}) = \sigma(k_1) + \xi(l). \] Recall [Muasugi 1970] that the invariant \( \xi(l) \) is defined by first orienting \( l \), giving, an oriented link \( \overline{l} \), say, and then setting \[ \xi(l) = \sigma(l) + Lk(\overline{k_1},\overline{k_2}) ,\] where \( \sigma \) denotes signature and \( Lk \) linking number.

In the present note we shall give an alternative, more conceptual, proof of Theorem 1, and in fact obtain it as a special case of a considerably more general result.

C. M. Gordon, R. A. Litherland, and K. Murasugi: “Signatures of covering links,” Can. J. Math. 33 : 2 (April 1981), pp. 381–394. MR 617628 Zbl 0469.57004 article

Richard Andrew Litherland	Related
Kunio Murasugi	Related

C. M. Gordon and R. A. Litherland: “Incompressible surfaces in branched coverings,” Chapter 7, pp. 139–152 in The Smith conjecture (New York, 6–7 April 1979). Edited by J. W. Morgan and H. Bass. Pure and Applied Mathematics 112. Academic Press (Orlando, FL), 1984. MR 758466 Zbl 0599.57006 incollection

Our primary aim is to show that if \( \widetilde{\Sigma} \) is the \( n \)-fold cyclic branched covering, \( n > 1 \), of a prime knot \( K \) in a homotopy 3-sphere \( \Sigma \), such that \( \Sigma - K \) is irreducible and contains a nonperipheral incompressible surface, then \( \widetilde{\Sigma} \) is not a homotopy sphere. This will be achieved by using the equivariant loop theorem to show that \( \widetilde{\Sigma} \) contains an incompressible surface of positive genus. Actually, we shall work in the more general context of a regular branched covering of a link in an arbitrary closed, orientable 3-manifold, as this is needed for the study of noncyclic finite group actions on homotopy 3-spheres. Our main result is Theorem 1, which asserts that, under certain circumstances, an incompressible surface in the complement of a link gives rise to one in any regular branched covering of the link. The special case of incompressible tori is considered in Theorem 2; here the proof actually uses the Smith conjecture. Interpreting Theorem 2 in terms of hyperbolic structures, using Thurston’s uniformization theorem for Haken manifolds (see Thurston [1979] and Chapter V, this volume by Morgan) we obtain Corollary 2.1, which states that if a regular branched covering of a link is hyperbolic, then so is the complement of the link.

These results are stated in Section 2, which also contains the necessary definitions and terminology. The proofs of Theorems 1 and 2 and Corollary 2.1 are given in Section 3. Finally, in Section 4, we give an elementary proof of the equivariant loop theorem for involutions (Theorem 3). A similar proof has been given by Kim and Tollefson [1980].

John Willard Morgan	Related
Richard Andrew Litherland	Related
Hyman Bass	Related

C. M. Gordon and R. A. Litherland: “Incompressible planar surfaces in 3-manifolds,” Topology Appl. 18 : 2–3 (December 1984), pp. 121–144. MR 769286 Zbl 0554.57010 article

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Cameron McAllan Gordon

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