In this paper we address the problem of finding the best viewpoints for three-dimensional straight-line graph drawings. We define goodness in terms of preserving the relational structure of the graph, and develop two continuous measures of goodness under orthographic parallel projection. We develop Voronoi variants to find the best viewpoints under these measures, and present results on the complexity of these diagrams.

We show that if a 3-dimensional polytopal complex has a knot in its 1-skeleton, where the bridge index of the knot is larger than the number of edges of the knot, then the complex is not constructible, and hence, not shellable. As an application we settle a conjecture of Hetyei concerning the shellability of cubical barycentric subdivisions of 3-spheres. We also obtain similar bounds concluding that a 3-sphere or 3-ball is non-shellable or not vertex decomposable. These two last bounds are sharp.

Abstract. We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of-1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction. 1.

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We report on a symbolic-numeric algorithm for computing the Alexander polynomial of each singularity of a plane complex algebraic curve defined by a polynomial with coefficients of limited accuracy, i.e. the coefficients are both exact and inexact data. We base the algorithm on combinatorial methods from knot theory which we combine with computational geometry algorithms in order to compute efficient and accurate results. Nonetheless the problem we are dealing with is ill-posed, in the sense that tiny perturbations in the coefficients of the defining polynomial cause huge errors in the computed results.

Temperley-Lieb algebras appear in exactly solvable statistical mechanics models as well as in the knot theory in mathematics. We have found many new solutions of Temperley-Lieb algebra in addition to some old known ones. We also discuss their rel- evance with respect to knot invariant. We have also found a class of solutions of the Yang-Baxter equation without the spectral parameter which satisfy a quadratic equation. PACS: 02.90.+p, 05.90+m 1.

© 2006 Darcy et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License

Necktie knots are inherently topological structures; what makes them tractable is the particular manner in which they are constructed. This observation motivates a map between tie knots and persistent walks on a triangular lattice. The topological structure embedded in a tie knot may be determined by appropriately manipulating its projection; we derive corresponding rules for tie knot sequences. We classify knots according to their size and shape and quantify the number of knots in a class. Aesthetic knots are characterised by the conditions of symmetry and balance. Of the 85 knots which may be tied with conventional tie, we recover the four traditional knots and introduce nine new aesthetic ones. For large (though impractical) half-winding number, we

Abstract. This paper will discuss different ways of finding rational parameterizations of knots in R3. We will first show a way of constructing torus knots, and then two ways of converting polynomial representations of knots into rational representations, the advantage of which is the possibility of one-point compactification in R3. This work was completed as part of the Mount Holyoke Summer Mathematics Institute, an NSF funded REU Program, under the advisement of Alan Durfee and Donal O’Shea. It was also completed while working with a research group consisting of