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

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

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

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

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: 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.

See also

• Slice genus
• Slice link
• Conway knot, a topologically slice knot whose smoothly non-slice status was unproven for 50 years