Slice knot

Slice knot

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 Ksubset 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.

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.


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