Stuck unknot
Encyclopedia
In mathematics
, a stuck unknot is a polygonal closed loop
that is topologically the unknot
but cannot be simplified to a planar
polygon
by rigid motions of the segments.
A related idea is to consider "stuck" polygonal open chains. Topologically these chains can be unknotted, but the limitation of using only rigid motions of the segments can create nontrivial
knots in such a chain.
Consideration of such "stuck" configurations arises in the study of molecular chains
in biochemistry
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a stuck unknot is a polygonal closed loop
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
that is topologically the unknot
Unknot
The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the...
but cannot be simplified to a planar
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...
polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
by rigid motions of the segments.
A related idea is to consider "stuck" polygonal open chains. Topologically these chains can be unknotted, but the limitation of using only rigid motions of the segments can create nontrivial
Nontrivial
Nontrivial is the opposite of trivial. In contexts where trivial has a formal meaning, nontrivial is its antonym.It is a term common among communities of engineers and mathematicians, to indicate a statement or theorem that is not obvious or easy to prove.-Examples:*In mathematics, it is often...
knots in such a chain.
Consideration of such "stuck" configurations arises in the study of molecular chains
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
in biochemistry
Biochemistry
Biochemistry, sometimes called biological chemistry, is the study of chemical processes in living organisms, including, but not limited to, living matter. Biochemistry governs all living organisms and living processes...
.