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A theory of signatures for odd-dimensional links in rational homology spheres are studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate 1-dimensional links with mutual linking. Furthermore our concordant theory of links in… (More)
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.
We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots — Naik's and Choi-Ko-Song's improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots — but are not… (More)
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants… (More)
A new approach to decoding of BCH and Reed-Solomon codes by using syzygy modules of matricies is introduced. The decoding procedure reduces the computational complexity roughly by half by introducing syzygy modules since the number of variables in each step is reduced by half. The decoding of binary BCH codes can be viewed as the special case of the… (More)
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.