897 research outputs found
Volume bounds for generalized twisted torus links
Twisted torus knots and links are given by twisting adjacent strands of a
torus link. They are geometrically simple and contain many examples of the
smallest volume hyperbolic knots. Many are also Lorenz links.
We study the geometry of twisted torus links and related generalizations. We
determine upper bounds on their hyperbolic volumes that depend only on the
number of strands being twisted. We exhibit a family of twisted torus knots for
which this upper bound is sharp, and another family with volumes approaching
infinity. Consequently, we show there exist twisted torus knots with
arbitrarily large braid index and yet bounded volume.Comment: Revised version to appear in Mathematical Research Letters. 21 pages,
14 figure
Compactifications of subvarieties of tori
We study compactifications of subvarieties of algebraic tori defined by
imposing a sufficiently fine polyhedral structure on their non-archimedean
amoebas. These compactifications have many nice properties, for example any k
boundary divisors intersect in codimension k. We consider some examples
including (and more generally log canonical models
of complements of hyperplane arrangements) and compact quotients of
Grassmannians by a maximal torus.Comment: 14 pages, submitted versio
Many projectively unique polytopes
We construct an infinite family of 4-polytopes whose realization spaces have
dimension smaller or equal to 96. This in particular settles a problem going
back to Legendre and Steinitz: whether and how the dimension of the realization
space of a polytope is determined/bounded by its f-vector.
From this, we derive an infinite family of combinatorially distinct
69-dimensional polytopes whose realization is unique up to projective
transformation. This answers a problem posed by Perles and Shephard in the
sixties. Moreover, our methods naturally lead to several interesting classes of
projectively unique polytopes, among them projectively unique polytopes
inscribed to the sphere.
The proofs rely on a novel construction technique for polytopes based on
solving Cauchy problems for discrete conjugate nets in S^d, a new
Alexandrov--van Heijenoort Theorem for manifolds with boundary and a
generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat
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