A slice knot is a
mathematical knot
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, (also known as ). Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of ...
in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
Definition
A knot
is said to be a topologically or smoothly slice knot, if it is the boundary of an
embedded disk in the 4-ball
, which is
locally flat or
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
, respectively. Here we use
: the
3-sphere is the
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
*Boundary (cricket), the edge of the pla ...
of the four-dimensional
ball Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology.
Smoothly slice knots are often illustrated using knots diagrams of
ribbon knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part o ...
s and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′).
Cone construction
The conditions locally-flat or smooth are essential in the definition: For every knot we can construct the
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
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).
Slice knots and the knot concordance group
Two oriented knots
are said to be concordant, if the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
is slice. In the same way as before, we distinguish topologically and smoothly concordant. With
we denote the
mirror image of
where in addition the orientation is reversed. The relationship ′concordant′ is reflexive because
is slice for every knot
. It is also possible to show that it is transitive: if
is concordant to
and
is concordant to
then
is concordant to
. Since the relation is also symmetric, it is an
equivalence relation. The equivalent classes together with the connected sum of knots as operation then form an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
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).
Examples
Every ribbon knot is a smoothly slice knot because—with the exception of the ribbon singularities—the knot already bounds an embedded disk (in 3-space). 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 knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s with crossing number
. These are
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
and
. Up to this crossing number there are no topologically slice knots which are not smoothly slice. Starting with crossing number 11 there is such an example, however: The
Conway knot
In mathematics, in particular in knot theory, the Conway knot (or Conway's knot) is a particular knot with 11 crossings, named after John Horton Conway.
It is related by mutation to the Kinoshita–Terasaka knot, with which it shares the sa ...
(named after
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...
) is a topologically but not smoothly slice knot. On the other hand, the Kinoshita-Terasaka knot, a so-called ′
mutant
In biology, and especially in genetics, a mutant is an organism or a new genetic character arising or resulting from an instance of mutation, which is generally an alteration of the DNA sequence of the genome or chromosome of an organism. It ...
′ of the Conway knot, is smoothly slice.
Twist knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite f ...
s are, except for the trivial knot and the
Stevedore knot
The stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot.
Naming
There is a lack of consensus among knot experts regarding the origin of t ...
, not slice. All topologically and smoothly slice knots with crossing number
are known.
Composite slice knots up to crossing number 12 are, besides those of the form
and
, the two more interesting knots
and
.
Invariants
The following properties are valid for topologically and smoothly slice knots:
The
Alexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a ve ...
of a slice knot can be written as
with a
Laurent polynomial
In mathematics, a Laurent polynomial (named
after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...
with integer coefficients (Fox-Milnor condition). It follows that the knot's determinant (
) is a square number.
The
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
is an invariant of concordance classes and the signature of slice knots is zero. Furthermore, the signature map is a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from concordance group to the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s: The signature of the sum of two concordance classes is the sum of the two signatures.
* It follows that the concordance group contains elements of infinite
order: The signature of a
trefoil knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest ...
is ±2 and the signature of the concordance class of the connected sum of
trefoils is
and therefore not 0.
* The concordance group also contains elements of order 2: The
figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...
is
amphicheiral and
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, and therefore we have
. In the concordance group we find
. Since the determinant of the figure-eight knot is 5, which is not a square number, this knot is not slice and it follows that its order in the concordance group is 2. Of course, knots with a finite order in the concordance group always have signature 0.
For both variants of the concordance group it is unknown whether elements of finite order
exist.
On the other hand, invariants with different properties for the two concordance variants exist:
Knots with trivial Alexander polynomial (
) are always topologically slice, but not necessarily smoothly slice (the Conway knot is an example for that). Rasmussen's s-invariant vanishes for smoothly slice, but in general not for topologically slice knots.
Geometrical description of the concordance relation
As an alternative to the above definition of concordance using slice knots there is also a second equivalent definition. Two oriented knots
and
are concordant if they are the boundary of a (locally flat or smooth) cylinder