Framing functions and strengthened version of Dehn's lemma

We give a lower estimate of the framing function of knots, and prove a strengthened version of Dehn's lemma conjectured by Greene-Wiest.Comment: 3 pages, 1 figure; More detailed argument adde

On the 3-dimensional invariant for cyclic contact branched coverings

We give a formula of 3-dimensional invariant for a cyclic contact branched covering of the standard contact S^{3}.Comment: 6 pages, 10 Figure

On a structure of random open books and closed braids

A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.Comment: 4 pages, no figure. Theorem 5 is adde

On a relation between the self-linking number and the braid index of closed braids in open books

We prove a generalization of the Jones-Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several counter examples for naive generalizations of the Jones-Kawamuro conjecture.Comment: 26 pages, 22 Figures; several minor errors correcte

On LMO invariant constraints for cosmetic surgery and other surgery problems for knots in S^{3}

We use the LMO invariant to find constraints for a knot to admit a purely or reflectively cosmetic surgery. We also get a constraint for knots to admit a Lens space surgery, and some information for characterizing slopes.Comment: 19 pages, many figure

A note on chirally cosmetic surgery on cable knots

We show that a $(p,q)$-cable of a non-trivial knot $K$ does not admit chirally cosmetic surgery for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that $(p,q)$-cable of non-trivial knot $K$ does not admit chirally cosmetic surgery as long as the JSJ piece of knot exterior does not contain $(2,r)$-torus exterior. We also show that an iterated torus knot other than $(2,p)$-torus knot does not admit chirally cosmetic surgery.Comment: 10 pages, 1 figur

Reading the dual Garside length of braids from homological and quantum representations

We show that Lawrence's representation and linear representations from quantum sl_2 called generic highest weight vectors detect the dual Garside length of braids in a simple and natural way. That is, by expressing a representation as a matrix over a Laurent polynomial ring using certain natural basis, the span of the variable is equal to the constant multiples of the dual Garside length.Comment: 18pages, 3 figure

Curve diagram for Artin group of type B

We develop a theory of curve diagrams for Artin groups of type B. We define the winding number labeling and the wall crossing labeling of curve diagrams, and show that these labelings detect the classical and the dual Garside length, respectively. A remarkable point is that our argument does not require Garside theory machinery like normal forms, and is more geometric in nature.Comment: 10 pages, 7 figures; modified to emphasize geometric aspect

Hurwitz equivalences of positive group generators

For a positively presented group G, we provide a criterion for two tuples of positive group generators of G to be Hurwitz equivalent or Hurwitz-conjugation equivalent. We also present an algorithmic approach to solve the Hurwitz equivalence and the Hurwitz search problems by using the word reversing method.Comment: 16 pages, 4 figures; errors in examples are corrected, some sections are re-writte

Space of group orderings, quasi morphisms and bounded cohomology

For a group $G$, we construct a quasi morphism from its left orderings and the map from the space of left orderings to the second bounded cohomology. We show that these maps reflect various properties of the group orderings.Comment: 14 pages, no figures, The statement of Theorem 1 is correcte