Double torus knot
Encyclopedia
A double torus knot is a closed curve drawn on the surface called a double torus
(think of the surface of two doughnut
s stuck together). More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus
two handlebody
in S³. If a link
is a subset of a genus two handlebody, it is a double torus link.
The simplest example of a double torus knot that is not a torus knot is the figure-eight knot
.
While torus knot
s and links are well understood and completely classified, there are many open questions about double torus knots.
Two different notations exist for describing double torus knots. The T/I notation is given in F. Norwood
, Curves on Surfaces and a different notation is given in P. Hill, On double-torus knots (I). The big problem, solved in the case of the torus, still open in the case of the double torus, is: when do two different notations describe the same knot?
Double torus
In mathematics, a genus-2 surface is a surface formed by the connected sum of two tori. That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified , forming a double torus.This is the simplest case of the connected sum of n tori...
(think of the surface of two doughnut
Doughnut
A doughnut or donut is a fried dough food and is popular in many countries and prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franchised specialty outlets...
s stuck together). More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
two handlebody
Handlebody
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...
in S³. If a link
Link (knot theory)
In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...
is a subset of a genus two handlebody, it is a double torus link.
The simplest example of a double torus knot that is not a torus knot is the figure-eight knot
Figure-eight knot (mathematics)
In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
.
While torus knot
Torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
s and links are well understood and completely classified, there are many open questions about double torus knots.
Two different notations exist for describing double torus knots. The T/I notation is given in F. Norwood
Rick Norwood
Rick Norwood is an American publisher, mathematician, comics historian and short story author.Born in Franklin, Louisiana, Norwood attended the Massachusetts Institute of Technology, where he was one of four writer-editors of the early underground comic God Comics, along with Bill Osten, Durk...
, Curves on Surfaces and a different notation is given in P. Hill, On double-torus knots (I). The big problem, solved in the case of the torus, still open in the case of the double torus, is: when do two different notations describe the same knot?