• Journal of the Korean Mathematical Society
• /
• v.52 no.6
• /
• pp.1305-1321
• /
• 2015

We dene new link invariants which are called the quasitoric braid index and the cyclic length of a link and show that the quasitoric braid index of link with k components is the product of k and the cycle length of link. Also, we give bounds of Gordian distance between the (p,q)-torus knot and the closure of a braid of two specific quasitoric braids which are called an alternating quasitoric braid and a blockwise alternating quasitoric braid. We give a method of modication which makes a quasitoric presentation from its braid presentation for a knot with braid index 3. By using a quasitoric presentation of $10_{139}$ and $10_{124}$, we can prove that $u(10_{139})=4$ and $d^{\times}(10_{124},K(3,13))=8$.

• Kyungpook Mathematical Journal
• /
• v.55 no.1
• /
• pp.191-203
• /
• 2015

We introduce a quasitoric virtual braid and show that every virtual link can be obtained by the closure of a quasitoric virtual braid. Also, we show that the set of quasitoric virtual braids with n strands forms a group which is a subgroup of the n-virtual braid group.

• Journal of the Korean Mathematical Society
• /
• v.54 no.3
• /
• pp.749-762
• /
• 2017

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands, then its two or respectively three dimensional representations are not faithful. We will also derive new presentations for the surface singular braid monoid, one with reduced the number of defining relations, and the other with reduced the number of its singular generators. We include surface singular braid formulations of all knotted surfaces in Yoshikawa's table.

• Kyungpook Mathematical Journal
• /
• v.55 no.1
• /
• pp.169-180
• /
• 2015

We study the braid indices of the Kanenobu knots. It is known that the Kanenobu knots have the same HOMFLYPT polynomial and the same Khovanov-Rozansky homology. The MFW inequality is known for giving a lower bound of the braid index of a link by applying the HOMFLYPT polynomial. Therefore, it is not easy to determine the braid indices of the Kanenobu knots. In our previous paper, we gave upper bounds and sharper lower bounds of the braid indices of the Kanenobu knots by applying the 2-cable version of the zeroth coefficient HOMFLYPT polynomial. In this paper, we give sharper upper bounds of the braid indices of the Kanenobu knots.

• Journal of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.673-693
• /
• 1997

In this work, the irreducible complex representations of degree 4 of $B_4$, the braid group on 4 strings, are classified. There are 4 families of representations: A two-parameter family of representations for which the image of $P_4$, the pure braid group on 4 strings, is abelian; two families of representations which are the composition of an irreducible representation of $B_3$, the braid group on 3 strings, with a certain special homomorphism $\pi : B_4 \longrightarrow B_3$; a family of representations which are the tensor product of 2 irreducible two-dimensional representations of $B_4$.

• Journal of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.179-193
• /
• 2003

The braid-permutation group is a group of welded braids which is the extension of Artin's braid groups by the symmetric groups. It is also described as a subgroup of the automorphism group of a free group. We also show that the plus-construction of the classifying space of the infinite braid-permutation group has the following two types of splittings BBP(equation omitted) B∑(equation omitted) $\times$ X, BBP(equation omitted) B $^{＋}$$\times$ Y＝S$^1$$\times$Y, where X, Y are some spaces.

• Honam Mathematical Journal
• /
• v.33 no.3
• /
• pp.407-418
• /
• 2011

There are two kinds of closing method for a given braid ${\beta}{\in}B_{2n}$, a braid closure $\hat{\beta}$ and a plat closure $\bar{\beta}$. In the article, we find a relation between the Conway potential function ${\nabla}_{\hat{\beta}}$ of braid closure $\hat{\beta}$ and ${\nabla}_{\hat{\beta}}$ of plat closure $\bar{\beta}$.

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.3
• /
• pp.555-561
• /
• 2010

For the pure braid groups, this paper gives a positive finite presentation whose generators are the standard Artin generators, and determines whether the submonoid generated by the Artin generators is a Garside monoid or not.

• Fibers and Polymers
• /
• v.5 no.3
• /
• pp.182-186
• /
• 2004

The effect of braid construction parameters on yarn cross-sectional shape is presented in this paper. The location of the yam within the braid unit cell is quantified by a compaction factor. A range of braided fabrics were produced and optically measured for actual yarn cross-sectional shape. A comparison of the theoretical and experimental values shows good correlation. Design curves can be produced with the developed model to allow selection of appropriate braid process parameter to create yarns with desired cross-sectional geometries.

• Journal of Adhesion and Interface
• /
• v.8 no.4
• /
• pp.14-22
• /
• 2007

In the present study, the effect of milled glass powder and liquid-type nitrile rubber (NBR) on the thermal stability of phenolic resin and the dynamic mechanical properties of glass braid/phenolic composites has been investigated by means of thermogravimetric analysis and dynamical mechanical analysis. It was found that both milled glass power and NBR filled in the waterborne phenolic resin significantly influenced the thermal stability of phenolic resins and the storage modulus and tan delta of the composites. The presence of glass powder increased the thermal stability of the phenolic resin, whereas the presence of NBR resulted in the weight loss in the specific temperature range. The thermal stability of the phenolic resins without and with the fillers was dependent not only on the cure temperature but also on the cure time. The variation of the storage modulus and tan ${\delta}$ of strip-type glass braid/phenolic composites was also influenced with the introduction of glass powder and NBR to the phenolic matrix as well as by the cure conditions given.