Author Topic: The same knot in two forms  (Read 342 times)

struktor

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The same knot in two forms
« on: May 26, 2019, 10:49:24 PM »
Does the name THK 3B4L apply in both cases?

Groundline

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Re: The same knot in two forms
« Reply #1 on: May 30, 2019, 02:00:27 AM »
No, the right diagram is a cylinder; the left is a mat.

struktor

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Re: The same knot in two forms
« Reply #2 on: May 30, 2019, 12:45:08 PM »
9 facet globe knot (4L3B).
Two points of view.

Stagehand

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Re: The same knot in two forms
« Reply #3 on: May 30, 2019, 08:53:55 PM »
Thank you Struktor, for your question and Groundline, for your reply. Yes, I say, they are the same knot, simply presented differently.  Struktor successfully demonstrates this by mapping the knot crossings and showing they may have the same tying order and may have the same projection.  This knot invariant is known in the math literature as Conway 9* and Rolfsen 9_40.   Groundline is not correct in distinguishing them.   As presented they are both mats and they are also both utilizable for several applications such as, as fixed loops or stoppers. Seeing knots as cylinders when they may be used as cylinders obscures their more general character as polyhedral and geometric.   Struktor further shows that both tying diagrams result in a single piece of geometry.  This knot?s geometry may have several names including 1)a truncated triangular prism, 2)an elongated triangular antiprism, 3)a square antiprism with an added vertex or knot crossing. This knot is simply the completed circuit of the True Carrick Bend.

struktor

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Re: The same knot in two forms
« Reply #4 on: May 31, 2019, 01:08:25 PM »
Thank you Stagehand, for interesting information.

Elongated triangular bipyramid  (or dipyramid).
6 rectangular triangles and 3 squares.

The cord is stable on this polyhedron.
Because  the knot is a geodesic loop.
« Last Edit: May 31, 2019, 01:57:19 PM by struktor »