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

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

## Definitions[]

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

$S^{3}=\{\mathbf {x} \in \mathbb {R} ^{4}:|\mathbf {x} |=1\}.$ The 3-sphere can be thought of as the boundary of the four-dimensional ball

$B^{4}=\{\mathbf {x} \in \mathbb {R} ^{4}:|\mathbf {x} |\leq 1\}.$ 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: 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, $8_{8}$ , $8_{9}$ , $8_{20}$ , $9_{27}$ , $9_{41}$ , $9_{46}$ , $10_{3}$ , $10_{22}$ , $10_{35}$ , $10_{42}$ , $10_{48}$ , $10_{75}$ , $10_{87}$ , $10_{99}$ , $10_{123}$ , $10_{129}$ , $10_{137}$ , $10_{140}$ , $10_{153}$ and $10_{155}$ . 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.

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.