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38
Finding the Best Viewpoints for ThreeDimensional Graph Drawings
 Proc. 5th International Symp. on Graph Drawing (GD ’97
, 1997
"... In this paper we address the problem of finding the best viewpoints for threedimensional straightline 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 ..."
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In this paper we address the problem of finding the best viewpoints for threedimensional straightline 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.
Nonconstructible complexes and the bridge index
, 1999
"... We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, 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 shella ..."
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Cited by 13 (2 self)
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We show that if a 3dimensional polytopal complex has a knot in its 1skeleton, 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 3spheres. We also obtain similar bounds concluding that a 3sphere or 3ball is nonshellable or not vertex decomposable. These two last bounds are sharp.
An obstruction to embedding 4tangles in links
 J. Knot Theory Ramifications
, 1999
"... Abstract. We consider the ways in which a 4tangle 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 pr ..."
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Abstract. We consider the ways in which a 4tangle 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 twointeger invariant of 4tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction. 1.
A Stevedore’s Protein Knot
, 2010
"... Protein knots, mostly regarded as intriguing oddities, are gradually being recognized as significant structural motifs. Seven distinctly knotted folds have already been identified. It is by and large unclear how these exceptional structures actually fold, and only recently, experiments and simulatio ..."
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Protein knots, mostly regarded as intriguing oddities, are gradually being recognized as significant structural motifs. Seven distinctly knotted folds have already been identified. It is by and large unclear how these exceptional structures actually fold, and only recently, experiments and simulations have begun to shed some light on this issue. In checking the new protein structures submitted to the Protein Data Bank, we encountered the most complex and the smallest knots to date: A recently uncovered ahaloacid dehalogenase structure contains a knot with six crossings, a socalled Stevedore knot, in a projection onto a plane. The smallest protein knot is present in an as yet unclassified protein fragment that consists of only 92 amino acids. The topological complexity of the Stevedore knot presents a puzzle as to how it could possibly fold. To unravel this enigma, we performed folding simulations with a structurebased coarsegrained model and uncovered a possible mechanism by which the knot forms in a single loop flip.
A proof of the KauffmanHarary conjecture
 Algebr. Geom. Topol
"... We prove the Kauffman–Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every nontrivial Fox p–coloring of D will assign different colors to different arcs. 57M25 ..."
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We prove the Kauffman–Harary Conjecture, posed in 1999: given a reduced, alternating diagram D of a knot with prime determinant p, every nontrivial Fox p–coloring of D will assign different colors to different arcs. 57M25
The Complexity of Computing Nice Viewpoints of Objects in Space
 IN VISION GEOMETRY IX, PROC. SPIE INTERNATIONAL SYMPOSIUM ON OPTICAL SCIENCE AND TECHNOLOGY
, 2000
"... A polyhedral object in 3dimensional space is often well represented by a set of points and line segments that act as its features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of the object, relevant for some task, are clearly visible. Such a vie ..."
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A polyhedral object in 3dimensional space is often well represented by a set of points and line segments that act as its features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of the object, relevant for some task, are clearly visible. Such a view is often called a nondegenerate view or projection. In this paper we are concerned with computing nondegenerate orthogonal and perspective projections of sets of points and line segments (objects) in 3dimensional space. We outline the areas in which such problems arize, discuss recent research on the computational complexity of these problems, illustrate the fundamental ideas used in the design of algorithms for computing nondegenerate projections, and provide pointers to the literature where the results can be found.
A SymbolicNumeric Algorithm for Computing the Alexander polynomial of . . .
"... We report on a symbolicnumeric 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 ..."
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We report on a symbolicnumeric 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 illposed, in the sense that tiny perturbations in the coefficients of the defining polynomial cause huge errors in the computed results.
ON THE HARARYKAUFFMAN CONJECTURE AND TURK’S HEAD KNOTS
, 2008
"... The m, n Turk’s Head Knot, THK(m, n), is an “alternating (m, n) torus knot.” We prove the HararyKauffman conjecture for all THK(m, n) except for the case where m≥5 is odd and n≥3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observ ..."
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The m, n Turk’s Head Knot, THK(m, n), is an “alternating (m, n) torus knot.” We prove the HararyKauffman conjecture for all THK(m, n) except for the case where m≥5 is odd and n≥3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m, 2) when Pm is a Pell prime.