Ambient isotopy

An object, such as a knot, or a link, or a graph, is topologically chiral if and only if it cannot be converted into its mirror image by continuous deformation (ambient isotopy) in the object s space otherwise it is topologically achiral. 

Listing conjectured that these two knot types cannot be transformed into one another by ambient isotopy. Figure 13 shows the two enantiomorphous isotopy types, arbitrarily designated as R and L. For each type, individual presentations are shown as projections in the plane, called diagrams. Crossings in these diagrams represent transverse double points, with over- and under-characteristics clearly... 

In summary, failure to detect a rigidly achiral presentation does not mean that such a presentation cannot be found among the infinitely many presentations of a knot failure to interconvert enantiomorphous presentations by ambient isotopy does not exclude the possibility that an interconversion pathway can be found among the infinitely many pathways that are available and a palindromic knot polynomial does not necessarily mean that the knot is amphicheiral. Consequently, it may be impossible in certain cases to determine with complete certainty whether a knot is topologically chiral or not. The fundamental task of the theory of knots was stated over a hundred years ago by its foremost pioneer Given the number of its double points, to find all the essentially different forms which a closed curve can assume. 15 Yet to find invariants that will definitively determine whether or not a knot is chiral remains an unsolved problem to this day.63a Vassiliev invariants have been conjectured to be such perfect invariants.63b... 

For the knot plane projection with defined passages, the following Reidemeister theorem is valid [39] different knots (or links) are topologically isomorphic to each other if they can be transformed continuously into one another by means of a sequence of simple local Reidemeister moves of types 1, 2 and 3 (see Fig. 9). Two knots are called regular isotopic if they are isomorphic with respect to the last two types of moves (2 and 3) if they are isomorphic with respect to all types of Reidemeister moves, they are called ambient isotopic. As can be seen from Fig. 9, a Reidemeister move of type 1 leads to the cusp creation on chain projection. At the same time, it is noteworthy that all real 3D-knots (links) are of ambient isotopy. 

The invariant of ambient isotopy of oriented knot or link is defined by ... 

Certain types of knots and links exist as topologically chiral enan-tiomorphs. Such enantiomorphs cannot be interconverted by continuous deformation ( ambient isotopy ). Homochirality classes can therefore be defined for this type of mathematical object. ... 

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