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My (undergrad) research group and I were working with this one specific class of knots, and we don't know quite how to search up their qualities to find out if they have a name.

Basically, they are knots with only one twist in their center, with some odd number x of crossings in their center twist. The trefoil is one such knot, with x=3. The next colorable one is at x=9. I don't think we're talking about Twist Knots, of which the Trefoil is also one such knot, because to my untrained eye the two look different. See below for x=9 "8-Bigon Twisty Knot" as we preliminarily called it:

knot with one twist and 9 crossings

So, do you know if these kinds of knots have a name? Or should we just call them "knot with one twist and x crossings"...

Thanks for your help!

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  • $\begingroup$ en.wikipedia.org/wiki/Pretzel_link $\endgroup$ Commented Mar 6, 2018 at 1:23
  • $\begingroup$ Ok, I was thinking of calling them (0,x,0) pretzel knots, but is that a thing? I thought for it to be classified as a pretzel knot the number of crossings in a tangle has to be more than 0 $\endgroup$ Commented Mar 6, 2018 at 1:26
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    $\begingroup$ The pictured knot is the $(2,9)$ torus knot. If there is one twist and $k$ crossings, then it is the $(2,k)$ torus knot (or link). With more than one twist region, perhaps it would be a pretzel knot or link, depending on how the twist regions are arranged. $\endgroup$ Commented Mar 6, 2018 at 1:30
  • $\begingroup$ Thank you Adam, that makes sense. I can see how I can redraw these as torus knots. $\endgroup$ Commented Mar 6, 2018 at 1:35

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This is the (9,2) torus knot, which you can draw this way:

(9,2) torus knot

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  • $\begingroup$ Thanks! This will help us in our midterm report and undergrad conference. $\endgroup$ Commented Mar 6, 2018 at 16:04

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