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Montesinos Amilibia, José María (1983) On twins in the foursphere. I. Quarterly Journal of Mathematics, 34 (134). pp. 171179. ISSN 00335606

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Official URL: http://qjmath.oxfordjournals.org/content/34/2/171.full.pdf+html
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http://qjmath.oxfordjournals.org/  Publisher 
Abstract
E. C. Zeeman [Trans. Amer. Math. Soc. 115 (1965), 471–495; MR0195085 (33 #3290)] introduced the process of twist spinning a 1knot to obtain a 2knot (in S4), and proved that a twistspun knot is fibered with finite cyclic structure group. R. A. Litherland [ibid. 250 (1979), 311–331; MR0530058 (80i:57015)] generalized twistspinning by performing during the spinning process rolling operations and other motions of the knot in threespace. The first paper generalizes those results by introducing the concept of a twin. A twin W is a subset of S4 made up of two 2knots R and S that intersect transversally in two points. The prototype of a twin is the ntwist spun of K (that is, the union of the ntwist spun knot of K and the boundary of the 3ball in which the original knot lies). The exterior of a twin, X(W), is the closure of S4−N(W), where N(W) is a regular neighborhood of W in S4.
The first paper considers properties of X(W), and uses these to characterize the automorphisms of a 2torus standardly embedded in S4, which extend to S4, and also to prove that any homotopy sphere obtained by Dehn surgery on such a 2torus is the real S4.
The second paper is devoted to the fibration problem, i.e. given a twin in S4, try to understand what surgeries in W give a twin W′ which has a component that is a fibered knot (as in the Zeeman theorem). This approach yields alternative proofs of the twistspinning theorem of Zeeman, and of the rolltwist spinning results of Litherland. New fibered 2knots are produced through these methods.
Item Type:  Article 

Uncontrolled Keywords:  twins in the foursphere; twistspinning a oneknot; twoknot; rolling; ntwin; Dehnsurgeries; Gluck's homotopy sphere 
Subjects:  Sciences > Mathematics > Topology 
ID Code:  17187 
Deposited On:  23 Nov 2012 11:56 
Last Modified:  02 Mar 2020 09:09 
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