Junctions as knots

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Junctions as knots

Postby Keiji » Mon May 27, 2013 8:33 am

So I tasked myself this morning with designing a 6-way freeflow interchange, allowing all movements straight ahead, and 120 degrees left and right. Sharp turns aren't allowed, because it's assumed traffic would have taken an earlier exit to get where they want (i.e. it would be doubling back). Here's what I came up with:

Image

Now, for a parclo or whirlpool interchange, it's pretty easy to look at the design and decide which road has to go over which other road in order to minimize the total vertical distance travelled (i.e. you don't want your roads to be going up over bridges then down under bridges then up over more bridges if you don't have to - you'd rather have them go over or under 3 bridges in a row and stay the same elevation throughout). But for a six way junction like this, it's a lot harder to decide which crossings have to be which way round. Now, I'm not an expert with knot theory, but I figured it could be useful here. Sadly, unlike junctions, knots don't have vertex-and-edge structures, they are just sets of closed loops. However, there is a fairly simple method to convert any vertex-and-edge structure into a set of closed loops - by treating each edge as a thin rectangle, then tracing around the sides. I've done this below:

Image

Do note that each end of the grey loop is supposed to be connected to the nearest other end where it leaves the image boundary, this just got cropped out. Now, although I haven't yet had any revelations on the most efficient way to choose the crossings, I was quite surprised by the symmetry of this knot. At first, I thought there would just be four loops - the grey one, and the red, green and blue ones. However, the entire reason I coloured them was to ensure all loops were accounted for - and I found exactly two more! When I saw the leftover edges, I was expecting three more for a total of seven, but no, there are a total of six. You have the grey loop, then the red, green and blue loops which are symmetric (up to crossing direction), and the yellow and cyan loops which are symmetric (again, up to crossing direction). I call this a 1+3+2 symmetry.

Has anyone else noticed this kind of knot symmetry between multiple loops in a single knot? I'm interested to find out what other possibilities there are.
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Keiji
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