Slice knot

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

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

A knot K ⊂ 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 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.

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.

The following is a list of all slice knots with 10 or fewer crossings; it was compiled using the Knot Atlas: 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 .