The splittability and triviality of 3-bridge links

Negami, Seiya; Okita, Kazuo

doi:10.1090/S0002-9947-1985-0779063-6

The splittability and triviality of $3$-bridge links
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by Seiya Negami and Kazuo Okita PDF

Trans. Amer. Math. Soc. 289 (1985), 253-280 Request permission

Abstract:

A method to simplify $3$-bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of $3$-bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every $3$-bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of $2$-fold branched coverings of ${S^3}$, it follows that every genus $2$ Heegaard diagram of ${S^2} \times {S^2}\# L(p,q)$ or ${S^3}$ can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.

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