16,712 research outputs found
The space of essential matrices as a Riemannian quotient manifold
The essential matrix, which encodes the epipolar constraint between points in two projective views,
is a cornerstone of modern computer vision. Previous works have proposed different characterizations
of the space of essential matrices as a Riemannian manifold. However, they either do not consider the
symmetric role played by the two views, or do not fully take into account the geometric peculiarities
of the epipolar constraint. We address these limitations with a characterization as a quotient manifold
which can be easily interpreted in terms of camera poses. While our main focus in on theoretical
aspects, we include applications to optimization problems in computer vision.This work was supported by grants NSF-IIP-0742304, NSF-OIA-1028009, ARL MAST-CTA W911NF-08-2-0004, and ARL RCTA W911NF-10-2-0016, NSF-DGE-0966142, and NSF-IIS-1317788
A class of knots with simple representations
We call a knot in the 3-sphere -simple if all representations of the
fundamental group of its complement which map a meridian to a trace-free
element in are binary dihedral. This is a generalisation of being a
2-bridge knot. Pretzel knots with bridge number are not
-simple. We provide an infinite family of knots with bridge number
which are -simple.
One expects the instanton knot Floer homology of a
-simple knot to be as small as it can be -- of rank equal to the knot
determinant . In fact, the complex underlying is of
rank equal to , provided a genericity assumption holds that is
reasonable to expect. Thus formally there is a resemblance to strong L-spaces
in Heegaard Floer homology. For the class of -simple knots that we
introduce this formal resemblance is reflected topologically: The branched
double covers of these knots are strong L-spaces. In fact, somewhat
surprisingly, these knots are alternating. However, the Conway spheres are
hidden in any alternating diagram.
With the methods we use, we show that an integer homology 3-sphere which is a
graph manifold always admits irreducible representations of its fundamental
group.Comment: 22 pages, 10 figures, to appear in Selecta Mathematic
Orders of elements in finite quotients of Kleinian groups
A positive integer will be called a {\it finitistic order} for an element
of a group if there exist a finite group and a
homomorphism such that has order in . It is
shown that up to conjugacy, all but finitely many elements of a given finitely
generated, torsion-free Kleinian group admit a given integer as a
finitistic order.Comment: 21 pp. I have largely rewritten Section 2 in order to correct the
statement of Proposition 2.7. The original statement was not logically clear,
and was not well adapted to an application in the more recent paper [22
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