Notice that the rule at each crossing can be stated as:
(part on the left) = (overpass) (part on the right)(overpass) -1
So from the relations at each crossing we have the following rules on the first line and then some deductions:
u = x y x -1,
x = y w y -1,
y = w z w -1,
w = z u z -1,
z = u x u -1.
y = wzw -1
= zuz -1 z zu -1z -1
= zuzu -1z -1
= uxu -1 u uxu -1 u -1 u x -1u -1
= uxuxu -1x -1u -1
= xyx -1 x xyx -1 x xy -1x -1 x -1 x y -1x -1
= xyxyxy -1x -1y -1x -1.
So xyxyxy -1x -1y -1x -1y -1 = 1.
Notice that in finding this rule we use four of the relations (one of the relations may be deduced from the others). Thus to take the loop off the knot, we must pass it through at least four of the crossings.
Now you should do a similar, and easier, exercise for the trefoil knot.
There is a lot more to be said in this area: to explain it all properly we need to develop the subject of `combinatorial group theory'.
In the above it is interesting to notice how we use symbols in different ways. Thus x denotes at one point a portion of a knot, then a piece of string winding around this part of the knot, then x is purely a symbol which we manipulate according to rules which model the geometry. Thus the symbols x, y, ... , and the way we combine them, also allow scope for analogy. This is how the modelling of geometry by algebra works. Of course the hard part is to find these rules, these laws, these means of modelling geometry by algebra, so leading to computation.
| Invariants | Paths and Loops |
© Mathematics and Knots, U.C.N.W.,Bangor, 1996 - 2002
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