Slice knot


A slice knot is a type of mathematical knot.

Definitions

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.
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, 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 :
61,,,,,,,,,,,,,,,,,,, and.

Properties

Every ribbon knot is smoothly slice.
An old question of Fox asks whether every smoothly slice knot is actually a ribbon knot.
The signature of a slice knot is zero.
The Alexander polynomial of a slice knot factors as a product where is some integral Laurent polynomial. This is known as the Fox–Milnor condition.