mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop. Isotopy for loops and quasigroups...

the other Isotopy of quasigroups. See Quasigroup#Homotopy and isotopy. Isotopy of loops, a triple of maps with certain properties. Isotopy of an algebra...

This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For a = b = c this is...

method of modifying the product on a Jordan algebra An isotope of an algebra: see Isotopy of algebras An isotope of a loop or quasigroup: see Isotopy of loops...

a "pointed idempotent". (That is, there is a principal isotopy (x, y, z) ↦ (x, −y, z).) A loop that is associative is a group. A group can have a strictly...

to the definition of isotopy. An ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the...

six loops, only one of which is a group, the cyclic group of order five. Below are two reduced Latin squares of order five, one from each isotopy class...

self-intersections simply by isotoping M into itself (the isotopy being in the domain of f), to a submanifold of M that does not contain the double-points. Thus...

deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted...

an isotopy. If the three maps of an isotopy are in S O ( 8 ) {\displaystyle \operatorname {SO(8)} } , the isotopy is called an orthogonal isotopy. If...

of uncountably many embeddings of a disk or sphere into three-dimensional space, all inequivalent in terms of ambient isotopy. Cantor dust – Set of points...

automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup. The isotopy group of the octonions...

the unlink. Although this construction of the knot treats its two loops differently from each other, the two loops are topologically symmetric: it is possible...

obtained by taking the connected sum of two identical trefoil knots 02 1 link/Unlink - equivalent under ambient isotopy to finitely many disjoint circles...

compact. Two knots are defined to be equivalent if there is an ambient isotopy between them. A knot in R3 (or alternatively in the 3-sphere, S3), can...

planar isotopy, can be related by a sequence of the three Reidemeister moves. Each move operates on a small region of the diagram and is one of three types:...

1, 2, 3 or 6 isotopy classes. The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflection...

classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid...

deformation of R 3 {\displaystyle \mathbb {R} ^{3}} upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted...

considered equivalent if there is an ambient isotopy of one tangle to the other keeping the boundary of the 3-ball fixed. Tangle theory can be considered...

are said to be equivalent if there is a continuous deformation of space (an ambient isotopy) taking one to another, and the Borromean rings may refer to...

Physics is a branch of science whose primary objects of study are matter and energy. Discoveries of physics find applications throughout the natural sciences...

the (universal cover of the) free loop space of a symplectic manifold. SFH is invariant under Hamiltonian isotopy of the symplectomorphism. Here, nondegeneracy...

two main deficiencies of CNF, one is that a continuous flow must be a homeomorphism, thus preserve orientation and ambient isotopy (for example, it's impossible...

natural notion of equivalence is isotopy. Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h-cobordisms...

an isotopy group over the real numbers that is spin group Spin(8), the unique such group that exhibits a phenomenon of triality. As a double cover of special...

polygonal loops. Erasing these loops, results in a polygonal curve without self-intersections which still lies close to the curve; some of its vertices...

deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting...

the path integral for the Wilson loops, we substitute for a source describing two particles moving in closed loops, i.e. J = J 1 + J 2 {\displaystyle...

3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by...