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Slice Knot


A knot K in S^3=partialD^4 is a slice knot if it bounds a disk Delta^2 in D^4 which has a tubular neighborhood Delta^2×D^2 whose intersection with S^3 is a tubular neighborhood K×D^2 for K.

Every ribbon knot is a slice knot, and it is conjectured that every slice knot is a ribbon knot.

The knot determinant of a slice knot is a square number (Rolfsen 1976, p. 224).

Unknot
SquareKnot

Slice knots include the unknot (Rolfsen 1976, p. 226), square knot (Rolfsen 1976, p. 220), stevedore's knot 6_1, and 9_(46) (Rolfsen 1976, p. 225), illustrated above.

Casson and Gordon (1975) showed that the unknot and stevedore's knot are the only twist knots that are slice knots (Rolfsen 1976, p. 226).


See also

Ribbon Knot, Tubular Neighborhood, Twist Knot

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References

Casson, J. and Gordon, C. M. "Cobordism of Classical Knots." Preprint of Lecture. Orsay. 1975.Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 218-220, 1976.

Referenced on Wolfram|Alpha

Slice Knot

Cite this as:

Weisstein, Eric W. "Slice Knot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SliceKnot.html

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