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I would like to extend the list of diagrams in the paper 'On strongly invertible knots' by Makoto Sakuma (1986) to knots with 10 crossings and succeeded for all but 8 of them: $10_{49}$, $10_{62}$, $10_{65}$, $10_{112}$, $10_{113}$, $10_{143}$, $10_{152}$, $10_{154}$.

Does anyone know of a method to obtain a symmetric diagram for these knots (or of an already existing list for 10 crossing knots)?

Some more detail: I look for transvergent diagrams for these 8 knots. Rotation about the axis then transforms the diagram into itself but with reversed orientation. If two axes are possible it would be great to have diagrams showing both axes simultaneously but for the moment I am fine with having diagrams with only one of them.

Edit, 24.05.2022: For $10_{112}$ the KLO diagram (see Marc's answer) based on the DT-code taken from Knot Info (and also the Knotscape diagram there) is already in the symmetric form (from the Rolfsen diagram it cannot be easily seen). For $10_{62}$ and $10_{65}$ I tried to modify the diagrams in KLO but was not yet successful.

enter image description here

Edit, 28.05.2022: Thanks Marc, for finding symmetric diagrams in all cases. For $10_{154}$ I transformed the intravergent diagram into the following transvergent one:

enter image description here

and the similar case of $10_{152}$ is obtained from this diagram by switching the crossing on the axis.

Therefore my goal to find symmetric diagrams for all strongly invertible knots with 10 crossings is achieved. In this set there are 45 2-bridge knots (these are always strongly invertible). There are 87 strongly invertible prime 3-bridge knots with 10 crossings (if I did not make an error in counting). I used the fact that invertible hyperbolic knots are strongly invertible and the symmetry information in Knot Info ('reversible' and 'fully amphicheiral' knots are invertible; note that there are several conventions for naming symmetries). I would like to present all 132 diagrams in a template fashion similar to the diagrams of symmetric unions in my paper "The search for nonsymmetric ribbon knots". One aim of this study is a comparison of these two cases ('symmetric' and 'anti-symmetric' diagrams; the first family denoting symmetric unions and the second strongly invertible knots with transvergent diagrams).

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  • $\begingroup$ I don't know whether this is what you want, but there are diagrams for all ten-crossing knots at katlas.org/wiki/The_Rolfsen_Knot_Table $\endgroup$
    – Gerry Myerson
    Commented May 23, 2022 at 5:01
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    $\begingroup$ Yes - and these 'standard' diagrams sometimes already show the required symmetry, as for instance in the case of $10_{116}$ (with a horizontal axis). But this is not the case for all 10 crossing knots which are strongly invertible. $\endgroup$
    – Christoph Lamm
    Commented May 23, 2022 at 5:56
  • $\begingroup$ My impression is usually the flow along the electrostatic potential gradient gets you pretty close to a maximal symmetry position, most of the time. I believe the Brakke surface evolver can do it: facstaff.susqu.edu/brakke/evolver/evolver.html $\endgroup$
    – Ryan Budney
    Commented May 24, 2022 at 22:10

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Here is a strongly invertible diagram of 10_49. The symmetry axis is almost horizontal.

enter image description here

This was created by loading the 'standard' diagram of 10_49 to KLO and then using KLO's simplifying method together with some work by hand. (KLO's simplifying method seems to prefer symmetric diagrams this is why I would expect the method to work in most cases.)

We have used similar methods for example here to create strongly invertible surgery diagrams. I would expect that the same works for the other 7 knots from your list. If you need help with that let me know.

EDIT: For the other knots it works similar. Here are diagrams of them.

10_62 10_62 10_65 10_65 10_113 10_113 10_143 10_143 10_152 10_152 10_154 10_154

(For the last two the symmetry axis is orthogonal to the projection plane.)

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  • $\begingroup$ Thanks, Marc, I will try that. Surprising that a simplifying method results in a 12 crossing diagram. I will add the solved cases in the comments. $\endgroup$
    – Christoph Lamm
    Commented May 24, 2022 at 5:35
  • $\begingroup$ Yes, you are right. Just pressing the 'Simplifying' bottom with the standard diagram will not change the diagram. First, one needs to modify the diagram by hand. $\endgroup$
    – Marc Kegel
    Commented May 24, 2022 at 7:49
  • $\begingroup$ I will go on experimenting with KLO but up to now I was not successful. I notice that it is possible to change the infinity-region and that there is a move based on an arc in the diagram (with two highlighted possibilities). However, I do not understand what its effect is and how to achieve R II moves. Finding the diagram for 10_112 was nice, at least. $\endgroup$
    – Christoph Lamm
    Commented May 24, 2022 at 19:47

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