#
EXAMPLES OF INVARIANTS:

##
THE UNKNOTTING NUMBER

Adams defines the unknotting number, another
invariant, like this:
**A knot K has unknotting number
n if there exists a projection of the knot such that changing n crossings
in the projection turns the knot into the unknot and there is no projection
such that fewer changes would have turned it into the unknot. [1]**

See
figure 15 for an example of this. I could not find a proof that the
unknotting number is in fact an invariant. Notice that this definition
assumes that there are a finite number of uncrossings that can be applied
to any knot to make it into the unknot. We will prove this using
results from Adams.

First
of all, we need to choose a starting point on the knot that is not at a
crossing, and start traveling in a particular direction along the knot.
Arriving at our first crossing, we observe whether the strand we are on
is the overstrand. If not, we change the crossing. We continue
in this fashion unless we reach a crossing where we have already been.
In this case, we leave the crossing as it is, and continue on what must
be the understrand. When we reach the starting point, we have produced
a knot which is easy to show is the unknot. See figure 16 for a visual
image of what was just described.

Applying an x-y-z-coordinate system to the
knot in 3-space with the z-axis coming straight out towards us, assume
that the initial point is at coordinate z = 1. Continuing to trace
along the knot, at each crossing we will decrease the z-coordinate at that
point, until the last point will have coordinate z = 0. But the last
point is the same as the first point, so we must put a vertical bar from
the last point to the first point to complete the knot. Now, as demonstrated
more clearly in figure 17, as we examine the knot straight down the z-axis,
the knot has no crossings. That is, the knot is the unknot.
Thus, we have shown that any arbitrary knot has a finite unknotting number.
[1]

**Next Example:** The
linking number

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