Narrow Results

A handlebody-knot is a handlebody embedded in the 3-sphere. We enumerate all genus two handlebody-knots up to six crossings.

A crossing change of a handlebody-knot is that of a spatial graph representing it. We see that any handlebody-knot can be deformed into trivial one by some crossing changes. So we define the unknotting numbers for handlebody-knots. In the case classical knots, which are considered as genus one handlebody-knots, Clark, Elhamdadi, Saito and Yeatman gave lower bounds of the Nakanishi indices by the numbers of some finite Alexander quandle colorings, and hence they also gave lower bounds of the unknotting numbers. In this paper, we give lower bounds of the unknotting numbers for handlebody-knots with any genus by the numbers of some finite Alexander quandle colorings of type at most $3$.

The unknotting number of a welded knot is considered. First, we obtain an upper-bound of the unknotting number of a welded knot by using the warping degree method. Further, we introduce a standard welded torus knot with welded datum and obtain an upper bound of the unknotting number by an algorithm with warping degree method. Secondly, we get a lower bound of the unknotting number of a welded knot by Alexander quandle colorings. Finally, we give a definition of Gordian distance for welded knots analogous to the classical case.

We give a simple condition for the existence of a nontrivial quandle coloring on a Montesinos link, which describes the distribution of the zeros of the Alexander polynomial. By this condition, we prove that the real parts of the zeros of the Alexander polynomial of any link that admits an alternating pretzel diagram are greater than $-1$: Hoste’s conjecture holds for such a link. Furthermore, we show the existence of infinitely many counterexamples for Hoste’s conjecture.

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal $n$-chart with two crossings I: Complementary domains of ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_{n-1}$, J. Knot Theory Ramifactions27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal $n$-chart with two crossings II: Neighbourhoods of ${\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_{n-1}$, Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math.113 (2019) 1693–1738, arXiv:1709.08827v2] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links.