We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specific Legendrian and transversal MTWS by enhancing the Legendrian mountain range for the (2,3)-cable of a (2,3)-torus knot provided by Etnyre and Honda, and showing that elementary negative...
General circle packings are arrangements of circles on a given surface such that no two circles overlap except at tangent points. In this paper, we examine the optimal arrangement of circles centered on concentric annuli, in what we term rings. Our motivation for this is two-fold: first, certain industrial applications of circle packing naturally allow for filled rings of circles; second, any packing of circles within a circle admits a ring structure if one allows for irregular spacing of circles...
The problem of utilizing facet reflections to bring a point outside of a convex polytope to inside has not been studied explicitly in the literature. Here we introduce two algorithms that complete the task in finite iterations. The first algorithm generates multiple solutions on the plane, and can be readily utilized in creating games on a plane or as a level generation method for video games. The second algorithm is a new efficient way to bring infeasible starting points of an optimization problem...
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