# Paper cutting (ConceptTopic, 4)

## Cutting in half

• A 0-twist (hose) goes to two separate 0-twists:
• A 2-twist goes to two linked together 2-twists:
• A 3-twist goes to a long 8-twist containing a trefoil knot:
• A 4-twist goes to two linked together 4-twists:
• A 5-twist goes to a long 12-twist containing a knot with five crossings:
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• A 6-twist goes to two linked together 6-twists:

## General rule

We can deduce that, when n > 0, cutting an n-twist will produce a single strip with n crossings and 2(n+1) twists if n is odd, or two linked strips each with n twists if n is even.

## Cutting in thirds

It is easy to see that cutting a loop into thirds rather than in half would be the same as above for an even number of twists. Therefore, the following concerns only loops with odd numbers of twists.

• A 1-twist goes to a short 1-twist linked to a long 4-twist:
• A 3-twist goes to a short 3-twist linked to a long 8-twist containing a trefoil knot:

(There are more crossings between the two loops than shown above)

## Repeated cuttings

• Cutting a 3-twist in half produces an 8-twist with a trefoil knot. Cutting this in half produces two interlinked 8-twists, each with their own trefoil knot, but also crossing in multiple other places.
• Cutting a 1-twist in thirds and then cutting the new 1-twist in half produces two 4-twists, linked in multiple places.