Slice knot

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

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

Definitions

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

${\displaystyle S^{3}=\{\mathbf {x} \in \mathbb {R} ^{4}\mid |\mathbf {x} |=1\}.}$

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

${\displaystyle B^{4}=\{\mathbf {x} \in \mathbb {R} ^{4}\mid |\mathbf {x} |\leq 1\}.}$

A knot ${\displaystyle K\subset S^{3}}$ 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, ${\displaystyle 8_{8}}$, ${\displaystyle 8_{9}}$, ${\displaystyle 8_{20}}$, ${\displaystyle 9_{27}}$, ${\displaystyle 9_{41}}$, ${\displaystyle 9_{46}}$, ${\displaystyle 10_{3}}$, ${\displaystyle 10_{22}}$, ${\displaystyle 10_{35}}$, ${\displaystyle 10_{42}}$, ${\displaystyle 10_{48}}$, ${\displaystyle 10_{75}}$, ${\displaystyle 10_{87}}$, ${\displaystyle 10_{99}}$, ${\displaystyle 10_{123}}$, ${\displaystyle 10_{129}}$, ${\displaystyle 10_{137}}$, ${\displaystyle 10_{140}}$, ${\displaystyle 10_{153}}$ and ${\displaystyle 10_{155}}$.[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 ${\displaystyle f(t)f(t^{-1})}$ where ${\displaystyle f(t)}$ is some integral Laurent polynomial.[4] This is known as the Fox–Milnor condition.[5]

See also

• Slice genus
• Slice link
• Conway knot, a topologically slice knot whose smoothly non-slice status was unproven for 50 years

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. 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.