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# Slice knot

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

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

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

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

A knot ${\displaystyle K\subset S^{3}}$ 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 ${\displaystyle f(t)f(t^{-1})}$ where ${\displaystyle f(t)}$ 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[full citation needed]: 61, ${\displaystyle 8_{8}}$, ${\displaystyle 8_{9}}$, ${\displaystyle 8_{20}}$, ${\displaystyle 9_{27}}$, ${\displaystyle 9_{41}}$, ${\displaystyle 9_{46}}$, ${\displaystyle 10_{3}}$, ${\displaystyle 10_{22}}$, ${\displaystyle 10_{35}}$, ${\displaystyle 10_{42}}$, ${\displaystyle 10_{48}}$, ${\displaystyle 10_{75}}$, ${\displaystyle 10_{87}}$, ${\displaystyle 10_{99}}$, ${\displaystyle 10_{123}}$, ${\displaystyle 10_{129}}$, ${\displaystyle 10_{137}}$, ${\displaystyle 10_{140}}$, ${\displaystyle 10_{153}}$ and ${\displaystyle 10_{155}}$.

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