Mark Gommers did indeed introduce the term “transposition”.
I’ve attached an image below showing the concept.
There are several IGKT posts over the years where a transposition has been submitted - but the various authors were not cognizant of the broader concept. I’ve also shown the “offset” orientation of this knot presented by “MasterTom”.
NOTE: All bends have corresponding ‘eye knots’ that can be derived from the parent bend. There are 4 linkages available - from which ‘eye knots’ can be derived. Some of these corresponding eye knots are homogenous to the parent bend (some aren’t). Four (4) eye knots can be derived from the available 4 linkages… and then a further 4 eye knots can be derived by way of transposition. This allows for up to 8 eye knots to be derived from a parent bend (when taking transposition into account).
NOTE: All of the derived eye knots will be tied within a particular chiral orientation which corresponds to the parent bends chirality. Of course, we can flip this to the opposite chirality. Tie any knot. Now hold it up to a plane mirror. The reflected image you see is its chiral opposite. The mirror of a knot can also be conceptualised as its inverse. Language is complicated, and that’s why I favour the term transposition instead of inverse.
Knot book authors generally only show a knot in one particular chiral orientation. For example, Ashley generally shows all of his ‘Bowlines’ in Z (right handed) chirality.
I am also curious as to why this wasn’t posted in the “New knots” category? It was offered as a potentially new knot.
