How do you define interlocked overhand bends?

Good question !

I was under the impression that I had answered it long ago - but just the day before yesterday I had read something that has awakened me. ( I hope it will be published soon, and then we will talk about it.)

So, till this day I was just considering the topology of each link of the bend. If each link/part was topologically equivalent to the overhand knot, the knot that was composed by those two links, inter-connected in no matter which way, I defined as an interlocked/interwoven/interpenetrating/interlinked overhand knots bend.

I started ... just from one overhand knot.

** NOW** I see your method !

( Well, at least I hope so...)

If you had just said : "

*I start from the one link of the bend, in the "usual / common" shape an overhand knot can have, and I see how the other link can be linked with this...*" , I would had understood it right from the start.

When you say :"

*I start from an overhand knot*..." , I am confused - bacause both links are overhand knots already- so how one can "start" and "finish" at the same point, without its path/argument be considered as cyclical ?

You speak of the "overhand knot" as

*a geometrical figure* -

**it is not**. On the contrary, I believe that this particular geometric shape of the overhand knot is responsible for great misunderstandings - like the current one !

I do not like this shape at all - unless it is resembling ( it is a portion of ) the symmetric shape of a trefoil knot - i.e. unless it has a three-fold point symmetry.

So, my current understanding of what you are saying is this : You start from the one link of the bend, in this particular shape an overhand not can hace, and you leave it unchanged/fixed till the very end. Then, you try to figure out how you can penetrate this link with the working end of the second link, keeping in mind that this second link should not get itself more entangled than an overhand knot.

I am afraid that this method, however simple and plausible might look at first sight, will not generate all the possible overhand knot-based bends. You will need some additional ideas, that will complicate it much more. Another disadvantage is that it can not distinguish, in advance, the symmetric from the non-symmetric knots. However, I can not be sure about its effectiveness or not - so I will wait and see the outcome of your efforts, with great interest.

Now, if you generate a knot that it is already well known, it would be better to label it with the already known name. If this method will produce

**all** the known overhand knot-based bends, then the labelling of all those knots with new names, indicative of the their particular place in the greater scheme where they will belong, according to this method, would make sense, indeed ( although, and unfortunately, the already given traditional name will not change - nobody calls oxygen by the numbers of protons, i.e, with their place in the periodic table of elements !

)