I am teaching a course on knots for the first time, and this led me to play with an approach which I have not seen in the literature. I would be surprised if no one had used it before, so I am asking whether this approach has a name, and if there are any relevant references.

The basic idea is like this. Let $\Lambda$ be the group freely generated by $\chi$ and $\rho$ subject to $\chi^2=\rho^4=1$. We will formulate everything in terms of finite sets with $\Lambda$-action. Let $U\subset\mathbb{R}^2\subset S^2$ be a link universe, i.e. a planar projection of a link, without any information about undercrossings or overcrossings (for the moment). Suppose for simplicity that there are no isolated circles. Let $A$ be the collection of pairs consisting of an arc in the diagram, together with a choice of direction for that arc. Define $\chi(a)$ to be the same arc with the opposite direction. Define $\rho(a)$ to be the arc obtained by "rotating $a$ through a quarter turn anticlockwise around the starting point", in an evident sense. This gives a $\Lambda$-action on $A$, and many of the standard things can be formulated in terms of this. The following three facts are enough to show that the combinatorial structure arises from a diagram in $S^2$:

  • The subgroup $\langle\rho\rangle$ acts freely
  • The stabiliser of every point is contained in the kernel of the map $c\colon\Lambda\to\{\pm 1\}$ given by $c(\chi)=c(\rho)=-1$.
  • If $B\subseteq A$ is a $\Lambda$-orbit with $|B|=n$ and $|B/\langle{\rho^{-1}\chi}\rangle|=m$ then $4m=n+8$. (This is because we can use $B$ to construct a CW structure on the ambient $S^2$, and then calculate its Euler characteristic.)

The components are the orbits for $\langle\rho^2\chi\rangle$, crossing information is given by a map $\omega\colon A\to\{\pm 1\}$ with $\omega\rho=-\omega$, an orientation is a map $\delta\colon A\to\{\pm 1\}$ with $\delta\rho^2=\delta\chi=-\delta$, a Reidemeister move of type $k\in\{1,2,3\}$ can be performed whenever you have an element $a$ with $(\rho^{-1}\chi)^k(a)=a$, a connected sum splitting is given by an element $a$ with $(\rho^{-1}\chi)^i(a)=(\rho\chi)^j(a)\neq a$ for some $i,j>0$, and so on.

  • 2
    $\begingroup$ This lo0ks like an approach to treating regular bipartite (insert a white vertex at the center of each edge) ribbon graphs as $\Lambda$-spaces, with appropriate group $\Lambda$. This is certainly well known. $\endgroup$ – Alex Degtyarev Feb 18 '15 at 9:45

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