The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state:
- Any reduced diagram of an alternating link has the fewest possible crossings.
- Any two reduced diagrams of the same alternating knot have the same writhe.
These were proven in 1987 and in 1991 correspondingly, using quantum topology. Reduced roughly means "no immediate Reidemeister 1 reductions", and alternating means crossings appear along link components in under-over-under-over order.
A w-link, sometimes called a welded link, is a virtual link in which one type of forbidden move is permitted, but not the other. A w-link is alternating if its real crossings appear in the alternating under-over-under-over pattern (i.e. ignore virtual crossings). I'd like to show that two alternating w-links are different based on the fact that they have reduced diagrams with different writhe, which is a Tait conjecture (I could compute invariants, but it feels like I shouldn't have to). Googling and searching "obvious sources" gave no information.
Question: What is the status of the corresponding Tait conjectures for alternating w-knots and w-links? Are they known, false, open...?
The statements are:
1. Any reduced diagram of an alternating w-link has the fewest possible real crossings.
2. Any two reduced diagrams of the same alternating w-knot have the same writhe.