Slice knot
Encyclopedia
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
and that the 3-sphere can be thought of as the boundary of the four-dimensional ball
A knot
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
where 
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:
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
and 
.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. It helps to remember that in knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a "knot" means an embedded circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
and that the 3-sphere can be thought of as the boundary of the four-dimensional ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
A knot
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
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
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
Ribbon knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disc with only ribbon singularities. This type of singularity is a self-intersection along an arc; the preimage of this arc consists of two arcs in the disc, one properly embedded in the disc and the...
is smoothly slice.
An old question of Fox
Ralph Fox
Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...
asks whether every slice knot is actually a ribbon knot.
The signature
Signature of a knot
The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface.Given a knot K in the 3-sphere, it has a Seifert surface S whose boundary is K...
of a slice knot is zero.
The Alexander polynomial of a slice knot factors as a product
where 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:
, , , , , , , , , , , , , , , , , , , and .