Narrow Results

A symmetric union of a knot is a generalization of the operation of the connected sum of a knot and its mirror image. In this paper, we show that every ribbon 2-knot of 1-fusion has a ribbon presentation associated with a symmetric union which has either the unknot or a 2-bridge knot as a partial knot.

We study the ribbon disks that arise from a symmetric union presentation of a ribbon knot. A natural notion of symmetric ribbon number r_S(K) is introduced and compared with the classical ribbon number r(K). We show that the difference r_S(K) - r(K) can be arbitrarily large by constructing an infinite family of ribbon knots K_n such that r(K_n) = 2 and r_S(K_n) > n. The proof is based on a particularly simple description of symmetric unions in terms of certain band diagrams which leads to an upper bound for the Heegaard genus of their branched double covers.

We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is equivalent to its corresponding knot. Second, under a different closure, the closed tangle is equivalent to the unlink. For each of these 21 ribbon knots, we present a 4-strand tangle that satisfies these properties. We provide diagrams of these tangles and also express them in planar diagram notation.

A symmetric union in the 3-space ${ℝ}^3$ is obtained from a knot in ${ℝ}^3$ and its mirror image, which is symmetric with respect to a 2-plane in ${ℝ}^3$, by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot $K$ is a composite symmetric union with minimal twisting number one, then $K$ has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two.

In this paper, we show that there exists a knot which has infinitely many non-equivalent symmetric union presentations with the same partial knots. We also define an invariant of a symmetric union presentation and give alternative proofs to results of Eisermann and Lamm and a result of Collari and Lisca by using methods in 3-dimensional topology.

We show that a two-bridge ribbon knot $K(m^2,mk\pm 1)$ with $m>k>0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon knots are also given.