KNOTS AND
TANGLES, ISOTOPY AND REIDEMEISTER
· When are two knots equivalent?
· What are the Reidemeister moves?
Q: What is a
knot?
A: A knot is a compact smooth 1-dimensional manifold embedded in R3. An equivalent definition is that of a piecewise linear, simple closed curve in R3. However, neither of these definitions gives an intuitive picture of what a knot is, so picture this instead: Imagine you have a rope, jumbled in some fashion. If you attach the two ends together to get a closed loop, you have a knot. A link is a finite number of knots that may or may not be intertwined.
Q: What
is a tangle?
A: There is a set (possibly empty) of
distinguished points in R2×{0} and another such set in R2×{1}. A tangle is a
set of disjoint smooth curves in R2×[0,1] that fall into one of the
following classes:
1)
If a curve is closed, then it lies in R2×(0,1)
2)
If the curve is not closed, then its
endpoints are in the union of the sets of distinguished points. Also, every distinguished point must be an
endpoint of some curve. The only points of
any curve that may lie in either boundary plane are the endpoints.
For an intuitive picture, imagine a set of coat hooks scattered in the ceiling, and another set of coat hooks in the floor. Now imagine strands of rope connecting the coat hooks. Each coat hook corresponds to exactly one rope. The ropes may go from ceiling to floor, or simply from ceiling to ceiling or floor to floor. In between, the ropes may be tangled around each other in any way. In addition, there may be closed loops of rope floating in the room and tangling with the other strands.
Clearly, the number of points in each distinguished set must have the same parity (either both odd or both even) since the total number of points must be even. Also, if both sets are empty, we are left simply with closed loops, i.e. links and knots. So knots and links are nothing more than special cases of tangles.
Q: When are two knots
equivalent?
A: Two
knots are equivalent if there is an isotopy between
them. An isotopy
is a continuous map with a continuous inverse.
Basically, imagine your closed loop of rope floating in space. Isotopy allows you
to stretch, bend, and twist the rope however you choose. The only thing you
cannot do are cut the rope and retie the ends.
Q: What are the Reidemeister moves?
A: Being
able to identify equivalent knots and tangles is rather important; yet, rarely
would you want to get a strand of rope and tie the knots you are interested
in. Rather, it is much easier to deal
with knot and tangle diagrams—two dimensional projections of knots and tangles
in which the “over” strand of each crossing is
marked. The above pictures are all
examples of knot diagrams.
The
Reidemeister moves are 3 operations on knot diagrams
which can transform a knot diagram into a diagram of any equivalent knot. These moves are:
(1) The twist
(2) The overlap
(3) The triple-point move
It is pretty obvious that these
moves represent isotopies on knots, but it is
somewhat surprising (and hence requires proof) that these are the only moves
necessary to represent any isotopy. We will not give a full proof here, but an
outline does give some intuition for what is really going on. It is fairly easy to believe that as you look
at an isotopy, you see that there are only two
important events: either a line segment crosses (over or under) a line segment, or a line segment crosses a crossing.
Line Segment Crossing Line
Segment: There are two cases, first, the segment could be crossing something
directly connected to it:
This takes care of the first Reidemeister move. The second case is if a line crosses one not
directly connected to it. In this case,
it must cross back at some point:
The second Reidemeister
move covers this scenario.
Line Segment Crossing a
Crossing: This breaks down into three subcases, each of which can be handled through combinations
of the second and third Reidemeister move:
This covers every case. Again, this is just an outline of a proof;
much detail would have to be added to make this rigorous.