Slice knot

A smooth slice disc in Morse position, showing minima, saddles and a maximum, and as an illustration a movie for the Kinoshita–Terasaka knot

A slice knot is a mathematical knot in 3-dimensional space that bounds a disc in 4-dimensional space.

Definitions[]

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

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: 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[]

The following is a list of all non-trivial slice knots with 10 or fewer crossings; 61, , , , , , , , , , , , , , , , , , , and .[2] All of them are smoothly slice.

Properties[]

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

References[]

  1. ^ Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 86, ISBN 9780387982540.
  2. ^ Livingston, C.; Moore, A.H., KnotInfo:Table of Knot Invariants
  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.