39 research outputs found
Homflypt Skein Modules.
Let k be a subring of the field of rational functions in x, v, s which contains x+/-1, v+/-1, s+/-1. If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Homflypt skein relations: (1) x --1L+ -- xL -- = (s -- s--1 )L0; (2) L with a positive twist = (xv--1)L; (3) L ⊔ O = u-u-1 s-s-1L where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the monomial basis of S( S1 x D2) given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for S(S 1 x D2) based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements s2n -- 1, for n a nonzero integer, and the elements s2m -- upsilon 2, for any integer m, are invertible in k, then S(S1 x S2) = k-torsion module ⊕ k. Here the free part is generated by the empty link &phis;. In addition, if the elements s2m -- upsilon 4, for m an integer, are invertible in k , then S(S1 x S2) has no torsion. We also obtain some results for more general k
A 2-chain can interlock with an open 10-chain
It is an open problem, posed in \cite{SoCG}, to determine the minimal
such that an open flexible -chain can interlock with a flexible 2-chain. It
was first established in \cite{GLOSZ} that there is an open 16-chain in a
trapezoid frame that achieves interlocking. This was subsequently improved in
\cite{GLOZ} to establish interlocking between a 2-chain and an open 11-chain.
Here we improve that result once more, establishing interlocking between a
2-chain and a 10-chain. We present arguments that indicate that 10 is likely
the minimum.Comment: 9 pages, 6 figure
The Homflypt skein module of a connected sum of 3-manifolds
If M is an oriented 3-manifold, let S(M) denote the Homflypt skein module of
M. We show that S(M_1 connect sum M_2) is isomorphic to S(M_1) tensor S(M_2)
modulo torsion. In fact, we show that S(M_1 connect sum M_2) is isomorphic to
S(M_1) tensot S(M_2) if we are working over a certain localized ring. We show
the similar result holds for relative skein modules. If M contains a separating
2-sphere, we give conditions under which certain relative skein modules of M
vanish over specified localized rings.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-31.abs.htm
Unsupervised Deep Epipolar Flow for Stationary or Dynamic Scenes
Unsupervised deep learning for optical flow computation has achieved
promising results. Most existing deep-net based methods rely on image
brightness consistency and local smoothness constraint to train the networks.
Their performance degrades at regions where repetitive textures or occlusions
occur. In this paper, we propose Deep Epipolar Flow, an unsupervised optical
flow method which incorporates global geometric constraints into network
learning. In particular, we investigate multiple ways of enforcing the epipolar
constraint in flow estimation. To alleviate a "chicken-and-egg" type of problem
encountered in dynamic scenes where multiple motions may be present, we propose
a low-rank constraint as well as a union-of-subspaces constraint for training.
Experimental results on various benchmarking datasets show that our method
achieves competitive performance compared with supervised methods and
outperforms state-of-the-art unsupervised deep-learning methods.Comment: CVPR 201
A 2-chain can interlock with a k-chain
One of the open problems posed in [3] is: what is the minimal number k such
that an open, flexible k-chain can interlock with a flexible 2-chain? In this
paper, we establish the assumption behind this problem, that there is indeed
some k that achieves interlocking. We prove that a flexible 2-chain can
interlock with a flexible, open 16-chain.Comment: 10 pages, 6 figure