Narrow Results

We resume work on telling embeddings of codimension two apart by counting colorings of the corresponding diagrams by given quandles. Previously, we illustrated the efficiency of this approach on classical knots. In the present paper we apply it to knotted surfaces. We recover work of Kamada in telling ribbon knots apart and we distinguish all elements of a class of twist-spun torus knots.

Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.

Let $K,K^{\prime }$ be ribbon knottings of $n$-spheres or tori in $S^{n+2}$, $n\ge 2$. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and Nakagawa, after taking a finite number of connected sums with trivially embedded copies of $S^{n-1}\times S^1$.

We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twist region, then it has a non-trivial knot and its mirror image as connected summands.