Slice knot

Examples

6₁

8₂₀

9₄₁

10₇₅

10₁₂₃

Ribbon knot

A slice knot is a type of mathematical knot.

Definitions

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

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

and that 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, 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

The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas: 6₁,[2] ${\displaystyle 8_{8}}$, ${\displaystyle 8_{9}}$, ${\displaystyle 8_{20}}$, ${\displaystyle 9_{27}}$, ${\displaystyle 9_{41}}$, ${\displaystyle 9_{46}}$, ${\displaystyle 10_{3}}$, ${\displaystyle 10_{22}}$, ${\displaystyle 10_{35}}$, ${\displaystyle 10_{42}}$, ${\displaystyle 10_{48}}$, ${\displaystyle 10_{75}}$, ${\displaystyle 10_{87}}$, ${\displaystyle 10_{99}}$, ${\displaystyle 10_{123}}$, ${\displaystyle 10_{129}}$, ${\displaystyle 10_{137}}$, ${\displaystyle 10_{140}}$, ${\displaystyle 10_{153}}$ and ${\displaystyle 10_{155}}$.

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

References

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. 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.