Narrow Results

We explain an algorithm for finding a boundary link Seifert matrix for a given multivariable Alexander polynomial. The algorithm depends on several choices and therefore makes it possible to find non-equivalent Seifert matrices for a given Alexander polynomial.

We characterize the Seifert matrices of periodic knots in S³ up to S-equivalence. Given a periodic knot we construct an equivariant spanning surface F and choose a basis for H₁(F) in such a way that the Seifert matrix has a special form exhibiting the periodicity. Conversely, given such a Seifert matrix we construct a periodic knot that realizes it. We exhibit the decomposition of H₁(F;ℂ) into eigenspaces of the periodic action, orthogonal to each other with respect to the Seifert pairing. Consequently we obtain Murasugi's formula for the Alexander polynomial of the periodic knot.

In this paper, we express the Seifert matrix of a periodic link which is presented as the closure of a 4-tangle with some extra restrictions, in terms of the Seifert matrix of the quotient link. As a result, we give formulae for the Alexander polynomial and the determinant of such a periodic link.

Using a special form of spanning surface for a knot, we give a formula for the coefficient of the $z^2$-term of the Alexander–Conway polynomial in terms of the sum of determinants of the blocks of $2\times 2$ submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

We show that for any positive integers $a,b,m,$ and $n$, the Alexander polynomial of the $(am,bn)$-Turk’s head link is divisible by that of the $(m,n)$-Turk’s head link.

Using Blanchfield pairings, we show that two Alexander polynomials cannot be realized by a pair of matrices with algebraic Gordian distance one if a corresponding quadratic equation does not have an integer solution. We also give an example of how our results help in calculating the Gordian distances, algebraic Gordian distances and polynomial distances.

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.

A double-torus knot is knot embedded in a genus two Heegaard surface in S³. After giving a notation for these knots, we consider double-torus knots L such that is not connected, and give a criterion for such knots to be non-trivial. Various new types of non-trivial knots with trivial Alexander polynomial are found.