Slice knot

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Examples
61
820
941
1075
10123
Ribbon knot

A slice knot is a type of mathematical knot.

Definitions[edit]

In knot theory, a "knot" means an embedded circle in the 3-sphere

and that the 3-sphere can be thought of as the boundary of the four-dimensional ball

A knot is slice if it bounds a nicely embedded 2-dimensional disk D in the 4-ball.[1]

What is meant by "nicely embedded" depends on the context, and there are different terms for different kinds of slice knots. If D is smoothly embedded in B4, then K is said to be smoothly slice. If D is only locally flat (which is weaker), then K is said to be topologically slice.

Examples[edit]

The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas[full citation needed]: 61,[2] , , , , , , , , , , , , , , , , , , and .

Properties[edit]

Every ribbon knot is smoothly slice. An old question of Fox asks whether every slice knot is actually a ribbon knot.[3]

The signature of a slice knot is zero.[4]

The Alexander polynomial of a slice knot factors as a product where is some integral Laurent polynomial.[4] This is known as the Fox–Milnor condition.[5]

See also[edit]

References[edit]

  1. ^ Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 86, ISBN 9780387982540.
  2. ^ "6 1", The Knot Atlas.
  3. ^ Gompf, Robert E.; Scharlemann, Martin; Thompson, Abigail (2010), "Fibered knots and potential counterexamples to the property 2R and slice-ribbon conjectures", Geometry & Topology, 14 (4): 2305–2347, arXiv:1103.1601, doi:10.2140/gt.2010.14.2305, MR 2740649.
  4. ^ a b Lickorish (1997), p. 90.
  5. ^ Banagl, Markus; Vogel, Denis (2010), The Mathematics of Knots: Theory and Application, Contributions in Mathematical and Computational Sciences, 1, Springer, p. 61, ISBN 9783642156373.