Slice knot

A slice knot is a type of mathematical knot. In knot theory, a "knot" means an embedded circle in the 3-sphere

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

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

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

A knot $K\subset S^{3}$ is slice if it bounds a nicely embedded 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.

Every ribbon knot is smoothly slice. An old question of Fox asks whether every slice knot is actually a ribbon knot.^[2]

The signature of a slice knot is zero.^[3]

The Alexander polynomial of a slice knot factors as a product $f(t)f(t^{{-1}})$ where $f(t)$ is some integral Laurent polynomial.^[3] This is known as the Fox–Milnor condition.^[4]

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

References

1. Lickorish, W. B. Raymond (1997), An Introduction to Knot Theory, Graduate Texts in Mathematics, 175, Springer, p. 86, ISBN 9780387982540.
2. 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, doi:10.2140/gt.2010.14.2305, MR 2740649.
3. Lickorish (1997), p. 90.
4. Banagl, Markus; Vogel, Denis (2010), The Mathematics of Knots: Theory and Application, Contributions in Mathematical and Computational Sciences, 1, Springer, p. 61, ISBN 9783642156373.

See also

• Slice genus
• Slice link