# 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

$S^{3}=\{\mathbf {x} \in \mathbb {R} ^{4}\mid |\mathbf {x} |=1\}$ 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 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 (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[full citation needed]: 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 $10_{155}$ .

## Properties

Every ribbon knot is smoothly slice. An old question of Fox asks whether every 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 $f(t)f(t^{-1})$ where $f(t)$ is some integral Laurent polynomial. This is known as the Fox–Milnor condition.