Twisted Alexander polynomials and a partial order on the set of prime knots

We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.Comment: This is the version published by Geometry & Topology Monographs on 25 February 200

Enumeration of Extractive Oracle Summaries

To analyze the limitations and the future directions of the extractive summarization paradigm, this paper proposes an Integer Linear Programming (ILP) formulation to obtain extractive oracle summaries in terms of ROUGE-N. We also propose an algorithm that enumerates all of the oracle summaries for a set of reference summaries to exploit F-measures that evaluate which system summaries contain how many sentences that are extracted as an oracle summary. Our experimental results obtained from Document Understanding Conference (DUC) corpora demonstrated the following: (1) room still exists to improve the performance of extractive summarization; (2) the F-measures derived from the enumerated oracle summaries have significantly stronger correlations with human judgment than those derived from single oracle summaries.Comment: 12 page

Twisted Alexander polynomials and surjectivity of a group homomorphism

If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-51.abs.htm

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.Comment: 33 pages, final version, to appear in Quantum Topolog