While thinking about this interesting question asked by Dylan Thurston, it occured to me that in every case that I know, the closer a knot diagram is to being alternating, the better its properties. For example, by the Tait Conjecture, a reduced diagram of an alternating link has the smallest number of crossings.
We can define the distance of a diagram from being alternating by a lexicographical ordering according to lengths of sequences of consecutive over/undercrossings. Other measures on non-alternating-ness are possible, but I'd like to ask about this "most naive" one.
- Is the crossing number of a knot type always realized by a `most alternating' reduced diagram according to the above metric? (This would hugely generalize the Tait Conjecture so surely the answer is no, but where can I find a counterexample?)
- And its stick number?
- And its prime decomposition? (related to this question.)
- And its representation of a braid closure/ plat closure?
Note that I'm not asking about diagrams in which various symmetries are visible, because then the statement would be false. For example, the following diagram for the figure eight knot illustrates its amphicheirality, but is not alternating:
Essentially, the question is whether the more alternating a diagram is, the better it is as a diagram. Does this basically seem to be the case?
To reverse the question and to make it easier to answer precisely, an equivalent question is:
Is there an example of "less alternating" reduced diagram with a better property as a diagram (excluding visualization of symmetries) than any "more alternating" diagram of the same knot?
Please feel free to replace the word "knot" by "link", "virtual knot", "virtual link", or even "virtual tangle" in the above question.