Some chapters are better than others. I couldn't really get into the DNA typography stuff, but I loved a lot of other sections of the book.

Looks like they combined two papers together.

I like the idea that virtual knot invariants extend to virtual tangles, as opposed to q-tangles.

Neat. I never noticed that section of Dror's website before! Do you happen to know which talks mention virtual knot theory?

Or, probably better, hyperbolic knots...

What if you restrict to prime knots?

Sorry, you should take pure braid generators to make this work! I will edit this to reflect that.

I never paused to consider whether $\pi_1(X)$ could be nonzero for non-Hausdorff finite spaces. Is there a simple way of describing your space $X$ as a quotient of $S^1$? You mentioned identifying the two hemispheres in some way!?