In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, especially in
knot theory
In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...
, a Wirtinger presentation is a finite
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
where the relations are of the form
where
is a word in the generators,
Wilhelm Wirtinger observed that the
complements of knots in
3-space have
fundamental groups with presentations of this form.
Preliminaries and definition
A ''
knot
A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
'' ''K'' is an embedding of the one-sphere ''S''
1 in three-dimensional space R
3. (Alternatively, the ambient space can also be taken to be the three-sphere ''S''
3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot,
is the knot complement. Its
fundamental group is an invariant of the knot in the sense that
equivalent knots have isomorphic
knot groups. It is therefore interesting to understand this group in an accessible way.
A ''Wirtinger presentation'' is derived from a regular projection of an
oriented knot. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing.
Wirtinger presentations of high-dimensional knots
More generally, co-dimension two
knots in
spheres are known to have Wirtinger presentations.
Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied:
# The
abelianization of the group is the integers.
# The 2nd
homology of the group is trivial.
# The group is finitely
presented.
# The group is the
normal closure of a single generator.
Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem.
Examples
For the
trefoil knot, a Wirtinger presentation can be shown to be
:
See also
*
Knot group
Further reading
* , section 3D
*
*
* {{Citation , last1=Livingston , first1=Charles , title=Knot Theory , year=1993, publisher=The Mathematical Association of America
Knot theory