Related papers: The Alexander polynomial of a plane curve singular…
We introduce a notion of tangential Alexander polynomials for plane curves and study the relation with $\theta$^Alexander polynomial. As an application, we use these polynomials to study a non-reduced degeneration $C_t \to D_0+jL$. We show…
Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r\vert G\vert$ variables of the algebraic link…
We give a formula for computing the characteristic polynomial for certain hyperplane arrangements in terms of the number of bipartite graphs of given rank and cardinality.
We describe symmetries of the braid monodromy decomposition for a class of plane curves defined over reals including the real curves with no real points and proving new divisibility relations for Alexander invariants of such curves.
In this paper, we compute Alexander polynomials of a torus curve C of type (2, 5), C : f(x, y) = f_2(x, y)^5 + f_5(x, y)^2 = 0, under the assumption that the origin O is the unique inner singularity and f2 = 0 is an irreducible conic. We…
Properties of the Alexander polynomials of Hurwitz curves are investigated. A complete description of the set of the Alexander polynomials of irreducible Hurwitz curves in the terms of their roots is given.
The Poincare series of a multi-index filtration on the ring of germs of functions can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring. Here this integral is considered with…
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
The augmentation variety of a knot is the locus, in the 3-dimensional coefficient space of the knot contact homology dg-algebra, where the algebra admits a unital chain map to the complex numbers. We explain how to express the Alexander…
In the present paper we compute Alexander polynomials for certain classes of conic-line arrangements in the complex projective plane which are related to pencils. We prove two general results for curve arrangements coming from Halphen…
In this paper, we characterize the polynomiality of surfaces of revolution by means of the polynomiality of an associated plane curve. In addition, if the surface of revolution is polynomial, we provide formulas for computing a polynomial…
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify…
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…
In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$…
We consider a braid $\beta$ which acts on a punctured plane. Then we construct a local system on this plane and find a homology cicle $D$ in its symmetric power such that $D\cdot \beta(D)$ coincides with the Alexander polynomial of the…
The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper we characterize the vanishing of such invariants for transversal unions of plane curves $C'$ and $C''$ in terms of the finiteness,…
We give a complete factorization of the invariant factors of resultant matrices built from birational parameterizations of rational plane curves in terms of the singular points of the curve and their multiplicity graph. This allows us to…
A univalent meromorphic function defined on $\Delta:= \{z \in \mathbb{C}: 1<|z|<\infty \}$ with univalent inverse defined on $\Delta$ is bi-univalent meromorphic in $\Delta$. For certain subclasses of meromorphic bi-univalent functions,…
We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of…
Let ${\cal C}$ be an algebraic space curve defined parametrically by ${\cal P}(t)\in {\Bbb K}(t)^{n},\,n\geq 2$. In this paper, we introduce a polynomial, the T--function, $T(s)$, which is defined by means of a univariate resultant…