A few days ago, my co-blogger Nathan Dunfield posted a counterexample to the Strong Neuwirth Conjecture.
N. Dunfield, A knot without a nonorientable essential spanning surface, arXiv:1509.06653
The Neuwirth Conjecture, posed by Neuwirth in 1963, asks roughly whether all knots can be embedded in surfaces in a way analogous to how a torus knot can be embedded in an unknotted torus. A weaker version, the “Weak Neuwirth Conjecture”, asks whether the knot group of any non-trivial knot in the 3-sphere can be presented as a product of free groups amalgameted along some subgroup. This was proven by Culler and Shalen in 1984. But nothing is proven about the ranks of these groups. The Neuwirth Conjecture would give the ranks as the genus of the surface. Thus, the Neuwirth Conjecture is an important conjecture for the structure theory of knot groups.
The Neuwirth Conjecture has been proven for many classes of knots, all via basically the same construction using a nonorientable essential spanning surface. The “Strong Neuwirth Conjecture” of Ozawa and Rubinstein asserts that this construction is always applicable because such a surface always exists.
Dunfield’s counterexample, verified by Snappea, indicates that we will need a different technique to prove the Neuwirth conjecture. Neuwirth’s Conjecture has just become even more alluring and interesting!