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Reidemeister-type moves for surfaces in four-dimensional space

Dennis Roseman — 1998

Banach Center Publications

We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in ℝ n + 2 (or S n + 2 ), for the cases n=2 or n=3. In a previous paper we have generalized the notion of the Reidemeister moves of classical knot theory. In this paper we examine in more detail the above mentioned dimensions. Examples are given; in particular we examine projections of twist-spun knots. Knot moves are given which demonstrate the triviality of the 1-twist spun trefoil. Another application is a smooth...

Elementary moves for higher dimensional knots

Dennis Roseman — 2004

Fundamenta Mathematicae

For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in ℝ n + 2 (or n + 2 ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

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