I hope "selfwrithe" is the established word. (0 for linkcrossing, otherwise identical to writhe +1 or 1) I bet the following is known: Take some crossing of a link with selfwrithe $w_a$. Flip it to get a link with $w_b$, call their arithmetic mean $w_{\times}$. Orient the crossing to overpass, split horizontally and vertically, respectively, to get links with selfwrithe $w_,w_$. The three numbers are linear dependent: $w_1w_2=w_2w_3$ (where ${\times,,}={1,2,3}$ but which is which depends on the selfwrithe of the crossing itself. It's nicely symmetric but I'm too idle to actually list the three subcases :) Thus I defined $J=w_1w_2$ (again with proper numbering, and a factor) to be the "angular momentum" of a crossing. E.g. Hopf link $J=1/2$, positive trefoil $J=+1$. You can do neat things with it: R1 crossings have $J=0$, and R2 pairs $+J,J$...too bad that one of the three crossings (the one that would make the R3 move pic alternating when flipped) of a R3 move changes J "randomly" or you could simply sum over all J of a link to get an invariant. Blast. (Still, I should go check now if there is a connection to the ThurstonBennequin number!)
Is there some use of the selfwrithe for knot polynomials (beyond the Kauffman bracket) ? (As usual, paper refs are welcome.)
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


