Related papers: The Alexander Polynomial of a Rational Link

This paper is an introduction to the state sum model for the Alexander-Conway polynomial that was introduced in the the author's book "Formal Knot Theory" (Princeton University Press, 1983). The article outlines how Alexander's original…

We provide a combinatorial characterisation of positive diagrams satisfying the equality in the Morton-Franks-Williams bound for the degrees of the HOMFLY-PT polynomial. This characterisation allows generating with relative ease examples of…

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize…

Fox's conjecture (1962) states that the sequence of absolute values of the coefficients of the Alexander polynomial of alternating links is trapezoidal. While the conjecture remains open in general, a number of special cases have been…

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then…

We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).

Alexander polynomial arises in the leading term of a semi-classical Melvin-Morton-Rozansky expansion of colored knot polynomials. In this work, following the opposite direction, we propose how to reconstruct colored HOMFLY-PT polynomials,…

We give an upper bound on the z-degree of the Kauffman polynomial of a link, using bridges of length greater than one which are separated in some tangle decomposition of a link diagram. We construct some examples by wiring together rational…

We show that the Alexander-Conway polynomial Delta is obtainable via a particular one-variable reduction of each two-variable Links-Gould invariant LG^{m,1}, where m is a positive integer. Thus there exist infinitely many two-variable…

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing…

The intersection index at a common point of two analytic varieties of complementary dimensions in $\Bbb C^n$ is positive. This observation, which has been called a ``cornerstone'' of algebraic geometry ([GH, p.~62]), is a simple consequence…

Murasugi discovered two criteria that must be satisfied by the Alexander polynomial of a periodic knot. We generalize these to the case of twisted Alexander polynomials. Examples demonstrate the application of these new criteria, including…

We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…

Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex Delta_{A,H} as the subdivision of the link of A induced by H. In particular, this generalizes Steingrimsson's coloring…

We introduce a new invariant of tangles along with an algebraic framework in which to understand it. We claim that the invariant contains the classical Alexander polynomial of knots and its multivariable extension to links. We argue that of…

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548,…

This note concerns a one-line diagrammatic proof of the Cayley-Hamilton Theorem. We discuss the proof's implications regarding the "core truth" of the theorem, and provide a generalization. We review the notation of trace diagrams and…

We show that the pattern $\{x,x+y,xy\}$ is partition regular over the space of formal integer polynomials of degree at least one with zero constant term, with primitive recursive bounds. This provides a new proof for the partition…

We show that there are links whose individual components are concordant to the unknot, but which are not concordant to any link with unknotted components. We give examples in the topological category, and examples in the smooth category…

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue…