Section 1.3
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Projections of Equivalent Knots
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Theorem: Given two projections of equivalent knots,
there is a finite sequence of Reidemeister moves
(defined below)and planar isotopies that take one
projection to the other.
Def. An ambient isotopy of R 3 is a continuous family
of homeomorphisms of R 3 . A planar isotopy of a knot
projection is a continuous deformation of the
projection plane.
Mth 333 – Spring 2013
Section 1.3
So, to show two projections represent the same knot,
it suffices to find a sequence of these moves that
takes one projection to the other.
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Reidemeister Moves
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Example
A Reidemeister move is one of three ways to change
a projection of a knot. The result is the projection of
an equivalent knot.
Example
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Amphichiral
Def: A knot is amphichiral if it is equivalent to its
mirror image.
The above example shows that the figure-eight knot
is amphichiral.
Mth 333 – Spring 2013
Section 1.3
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