Narrow Results

We construct new invariant polynomials in two and multiple variables for virtual knots and links. They are defined as determinants of Alexander-like matrices whose determinants are virtual link invariants. These polynomials vanish on classical links. In some cases, they separate links that can not be separated by the Jones–Kauffman polynomial [Kau] and the polynomial proposed in [Ma3].

In this note, we will give an approach to determine the unknotting number by a surgical view of Alexander matrix.

Let be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom(K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism

In the previous note, Okada and the author gave an approach to give a characterization of Alexander polynomials for knots which are transformed by a single crossing change into a given knot whose Alexander polynomial is monic. In this note, we give a characterization in the case of $8_{20}$, and show that the Gordian distance of $4_1$ and $8_{20}$ is two. We also give a characterization in the cases of more three knots.

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

In this paper, we find that for each given spatial graph, the number of its maximal trees is finite, and the quotient space corresponding to some maximal tree is also a spatial graph and it is homeomorphic to a disjoint union of wedge sum of circles. That means this quotient space is orientable and colorable. Thus, we define two equivalence invariants of spatial graphs, the coloring number bracket and the Alexander invariant set, which are coming from the coloring invariant and Alexander matrix.