Slice knot

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A slice knot is a type of mathematical knot. It helps to remember that 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 $K\subset \mathbb{S}^3$ is slice if it bounds a nicely embedded 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 K is only locally flat (which is weaker), then K is said to be topologically slice.

Any ribbon knot is smoothly slice. An old question of Fox asks whether every slice knot is actually a ribbon knot.