In 1965, E. C. Zeeman proved that the (±1)-twist spin of any knotted sphere in (n−1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman’s theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (±1)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (±1)-twist spins are always unknotted.

Original languageEnglish
Pages (from-to)1371-1382
Number of pages12
JournalProceedings of the American Mathematical Society
Issue number3
StatePublished - Mar 2016


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