A 2-chain can interlock with an open 10-chain

It is an open problem, posed in \cite{SoCG}, to determine the minimal $k$ such that an open flexible $k$-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

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

Banzhaf random forests: Cooperative game theory based random forests with consistency.

Random forests algorithms have been widely used in many classification and regression applications. However, the theory of random forests lags far behind their applications. In this paper, we propose a novel random forests classification algorithm based on cooperative game theory. The Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. Hence, we call the proposed algorithm Banzhaf random forests (BRFs). Unlike the previously used information gain ratio, which only measures the power of each feature for classification and pays less attention to the intrinsic structure of the feature variables, the Banzhaf power index can measure the importance of each feature by computing the dependency among the group of features. More importantly, we have proved the consistency of BRFs, which narrows the gap between the theory and applications of random forests. Extensive experiments on several UCI benchmark data sets and three real world applications show that BRFs perform significantly better than existing consistent random forests on classification accuracy, and better than or at least comparable with Breiman’s random forests, support vector machines (SVMs) and k-nearest neighbors (KNNs) classifiers