Slice knot
Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat.
Both types of slice knots are important in 3- and 4-dimensional topology.
The conditions locally-flat or smooth are essential in the definition: For every knot we can construct the cone over the knot which is a disk in the 4-ball with the required property with the exception that it is not locally-flat or smooth at the singularity (it works for the trivial knot, though).
Note, that the disk in the illustration on the right does not have self-intersections in 4-space.
These only occur in the projection to three-dimensional space.
Therefore, the disk is ′correctly′ embedded at every point but not at the singularity (it is not locally-flat there).
The equivalence classes together with the connected sum of knots as operation then form an abelian group which is called the (topological or smooth) knot concordance group.
The neutral element in this group is the set of slice knots (topological or smooth, respectively).
The ribbon singularities may be deformed in a small neighbourhood into 4-space so that the disk is embedded.
There are 21 non-trivial slice prime knots with crossing number
[1] Starting with crossing number 11 there is such an example, however: The Conway knot (named after John Horton Conway) is a topologically but not smoothly slice knot.
[3] All topologically and smoothly slice knots with crossing number
[4] Composite slice knots up to crossing number 12 are, besides those of the form
[5] The following properties are valid for topologically and smoothly slice knots: The Alexander polynomial of a slice knot can be written as
For both variants of the concordance group it is unknown whether elements of finite order
On the other hand, invariants with different properties for the two concordance variants exist: Knots with trivial Alexander polynomial (
are concordant if they are the boundary of a (locally flat or smooth) cylinder
with different orientations[8] and therefore two mirrored trefoils are shown as boundary of the cylinder.
Connecting the two knots by cutting out a strip from the cylinder yields a disk, showing that for all knots the connected sum
This can be illustrated also with the first figure at the top of this article: If a small disk at the local minimum on the bottom left is cut out then the boundary of the surface at this place is a trivial knot and the surface is a cylinder.
If the disk (or cylinder) is smoothly embedded it can be slightly deformed to a so-called Morse position.
This is useful because the critical points with respect to the radial function r carry geometrical meaning.
At saddle points, trivial components are added or destroyed (band moves, also called fusion and fission).
For slice knots any number of these band moves are possible, whereas for ribbon knots only fusions may occur and fissions are not allowed.
In the illustration on the right the geometrical description of the concordance is rotated by 90° and the parameter r is renamed to t. This name fits well to a time interpretation of a surface ′movie′.
An analogous definition as for slice knots may be done with surfaces of larger genus.
The 4-genus is always smaller or equal to the knot's genus because this invariant is defined using Seifert surfaces which are embedded already in three-dimensional space.
Examples for knots with different values for their topological and smooth 4-genus are listed in the following table.
Judging from the values in the table we could conclude that the smooth and the topological 4-genus always differ by 1, when they are not equal.
This is not the case, however, and the difference can be arbitrarily large.
