A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 of *An introduction to the theory of groups* by Rotman. This property means that a knot group is completely determined by the fundamental group of a Seifert surface complement, plus a choice of meridian. I wonder whether this is somehow the real reason that Seifert surfaces are important in knot theory. Morally, the fact that a knot group is an HNN extension means that all the information about a knot which you might care about is contained in any of its Seifert surfaces.

To remind you of the explicit group-theoretic statement, there is an isomorphism

$\phi\colon\thinspace \frac{\pi(F)\ast \langle m\rangle}{\mathcal{N}}\longrightarrow \pi(K),$

where $\langle m\rangle$ is the infinite cyclic group generated by the meridian, and $\mathcal{N}$ is the smallest normal subgroup of the free product $\pi(F)\ast \langle m\rangle$ containing the elements

$m^{-1}\mu^+(z)m(\mu^{-}(z))^{-1}\qquad z\in \pi(F),$

with $\mu^{\pm}$ denoting the pushoff maps.

There is a natural notion of a virtual knot group, by assigning a formal generator to each arc of a virtual knot diagram, and a Wirtinger relation to each real crossing (virtual crossings are ignored). Any Wirtinger presentation of deficiency $0$ or $1$ can be realized as a virtual knot group by Theorem 3 of a paper by Se-Goo Kim.

Question: Is every virtual knot group an HNN extension? (edit: over a finitely generated group?) Can the base group be described in terms of a group generated by a commutator at each real crossing? If not, is a virtual knot group "almost" an HNN extension in some useful sense?

I'm interested in this question because I wonder whether invariants coming from Seifert surfaces can be read off Gauss diagrams in any systematic way. Are Seifert surfaces an essential feature of knots, as opposed to virtual knots; or are they a non-essential luxury?

finitely presentedgroup $G$, every surjective homomorphism $f:G\to\mathbf{Z}$ arises from some splitting of $G$ as an HNN extensionover a finitely generated subgroup(of the kernel of $f$). $\endgroup$ – YCor Jun 14 '18 at 10:43