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# crossing change

In a knot, a crossing change occurs when a positive crossing (or crossing over) is followed by a negative crossing (or crossing under). An alternating knot, for example, has exactly as many crossing changes as it has crossings.

For the purpose of illustrating crossing changes, a non-alternating knot is more interesting. In the following diagram, if we start at the point indicated by the blue dot and proceed clockwise, the red arrows indicate crossing changes while the green arrows indicate a crossing the same as the previous crossing. One crossing change after that and there are no further arrows in the diagram until the blue dot.

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## Comments

## Weird knot universe

> when is a "crossing over" not a "crossing under"?

Does anyone know of some weird, exotic, theoretical universe in which a crossing over in a knot can also be a crossing under?

## Re: Weird knot universe

Does anyone know of some weird, exotic, theoretical universe in which a crossing over in a knot can also be a crossing under?

Our own universe. Sorry if that sounds like a smart-aleck response. For example, a kid who ties his shoes with double knots, or even triple knots. Then some crossings over will also be crossings under.

## Re: Weird knot universe

> Our own universe. Sorry if that sounds like a smart-aleck response. For example, a kid who ties his shoes with double knots, or even triple knots. Then some crossings over will also be crossings under.

So then are you saying the answer is: when each crossing involves only two subsections of the knot?

## Re: Weird knot universe

That would satisfy me, but I'm no super-literalist and I don't care much for theoretical universes (outside the context of sci-fi).