Narrow Results

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

There is a classical correspondence between edge-signed plane graphs and link diagrams. Determining component number of links corresponding to plane graphs may be one of the first problems in studying links by using graphs. There has been several early studies in this aspect, for example, the component number of links formed from 2-dimensional square lattices (4⁴) has been determined. In this paper, we determine the component number of links corresponding to 2-dimensional triangular (3⁶) and honeycomb (6³) lattices with free or cyclic boundary condition.

A short, elementary proof is given of the result that the number of components of a link arising from a medial graph M(Γ) by resolving vertices is equal to the nullity of the mod-2 Laplacian matrix of Γ.

In this note, we first give an alternative elementary proof of the relation between the determinant of a link and the spanning trees of the corresponding Tait graph. Then, we use this relation to give an extremely short, knot theoretical proof of a theorem due to Shank stating that a link has component number one if and only if the number of spanning trees of its Tait graph is odd.

If a set of local moves can transform every knot into a trivial knot, it is called a generalized unknotting operation. The author collects generalized unknotting operations and classify them up to local equivalence.