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Conversely, two knots equivalent under the ambient isotopy definition are also equivalent under the orientation-preserving homeomorphism definition, because the = (final) stage of the ambient isotopy must be an orientation-preserving homeomorphism carrying one knot to the other. The basic problem of knot theory, the recognition problem, is ...
A polygonal knot is a knot whose image in R 3 is the union of a finite set of line segments. [6] A tame knot is any knot equivalent to a polygonal knot. [6] [Note 2] Knots which are not tame are called wild, [7] and can have pathological behavior. [7] In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for ...
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3.
This idea is used in Kuperberg's result that the unknotting problem is in co-NP. Knot Floer homology of the knot detects the genus of the knot, which is 0 if and only if the knot is an unknot. A combinatorial version of knot Floer homology allows it to be computed (Manolescu, Ozsváth & Sarkar 2009).
No other knot with 10 or fewer crossings has trivial Alexander polynomial, but the Kinoshita–Terasaka knot and Conway knot (both of which have 11 crossings) have the same Alexander and Conway polynomials as the unknot. It is an open problem whether any non-trivial knot has the same Jones polynomial as the unknot.
The Kissing Number Problem. A broad category of problems in math are called the Sphere Packing Problems. They range from pure math to practical applications, generally putting math terminology to ...
In the branch of mathematics called knot theory, the volume conjecture is an open problem that relates quantum invariants of knots to the hyperbolic geometry of their complements. Statement [ edit ]
A college student just solved a seemingly paradoxical math problem—and the answer came from an incredibly unlikely place.