Ki Hyoung Ko

A new presentation of then-string braid groupBnis studied. Using it, a new solution to the word problem inBnis obtained which retains most of the desirable features of the Garside–Thurston solution, and at the same time makes possible certain computational improvements. We also give a related solution to the conjugacy problem, but the improvements in its complexity are not clear at this writing.

We propose a new digital signature scheme based on a non-commutative group where the conjugacy search problem is hard and the conjugacy decision problem is feasible. We implement our signature scheme in the braid groups and prove that an existential forgery of the implementation under no message attack gives a solution to a variation of conjugacy search problem. Then we discuss performance of our scheme under suggested parameters.

In this paper we study a combinatorial matrix considered by W. B.

To an m-component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link.An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant.

We classified the set of F m -cobordism classes of F m -links by their Seifert matrices in [ 5 ]. On the other hand Cappell and Shaneson identified them with essentially a quotient group of their homology surgery obstruction group [ 2 ]. In this paper, we will find a description of their surgery obstruction in terms of a Seifert matrix. In relation to Ledimet's recent results [ 7 ], we hope this might provide some clue to whether F m -cobordism or boundary cobordism is stronger than ordinary lin

Abstract. The framed braid group on n strands is defined to be a semidirectproduct of the braid group B„ and Z . Framed braids represent 3-manifoldsin a manner analogous to the representation of links by braids. Consider twoframed braids equivalent if they represent homeomorphic 3-manifolds. Themain result of this paper is a Markov type theorem giving moves that generatethis equivalence relation. In this paper the group of framed braids $n is introduced. This group issimilar to the braid group a

To an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link. An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by

The framed braid group on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> strands is defined to be a semidirect product of the braid group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <

We obtain the genera of knots or links that are the closures of certain types of 4-braids. These 4-braids are written as positive words in the band-generator presentation of the 4-braid group considered by E.S. Kang et al. in this volume, and their closures naturally bound surfaces that consist of four disks and half-twisted bands connecting any two disks. In fact we show that these spanning surfaces have the minimal genera.

We propose an algorithm for deciding whether a given braid is pseudo-Anosov, reducible, or periodic. The algorithm is based on Garside's weighted decomposition and is polynomial-time in the word-length of an input braid. Moreover, a reduction system of circles can be found completely if the input is a certain type of reducible braids.