Keiko Kawamuro

Essential open book foliations and fractional Dehn twist coefficient

We introduce essential open book foliations by refining open book foliations, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the ‘gap’ between overtwisted contact structures and non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen–Thurston types of the monodromies of their open book decompositions.

The algebraic crossing number and the braid index of knots and links

It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable of K and for the connect sum of K and L.

Open book foliation

We study open book foliations on surfaces in 3-manifolds, and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula of the self-linking number of transverse links, which reveals an unexpected link to the Johnson-Morita homomorphism in mapping class group theory. We also give an alternative combinatorial proof to the Bennequin-Eliashberg inequality.

VISUALIZING OVERTWISTED DISCS IN OPEN BOOKS

We give an alternative proof of a theorem of Honda-Kazez-Matic that every non-right-veering open book supports an overtwisted contact structure. We also study two types of examples that show how overtwisted discs are embedded relative to right-veering open books. In (15), we have introduced open book foliations and their basic machinery by using that of braid foliations (2, 3, 4, 5, 6, 7, 8, 9) and showed applications of open book foliations including a self-linking number formula of general closed braids. In (16) we study the geometric structure of a 3-manifold by using open book foliations. In this note we study more applications of open book foliations. We will assume the readers are familiar with the definition and basic machinery of open book foliations in (15). One of the features of open book foliations is that one can visualize how surfaces are embedded with respect to general open books. In this paper we use this feature to illustrate overtwisted discs and give constructive methods to detect overtwisted contact structures. We first give an alternative proof of a tightness criterion theorem by Honda, Kazez and Matic (14): If an open book is not right-veering then it supports an overtwisted contact structure. The converse does not hold: In fact, Honda, Kazez and Matic (14) show that if a contact structureis supported by a non-right veering open book (S,�), by applying positive stabilizations to (S,�) one can find a right-veering open book ( b S, b �) that supports �. We

Overtwisted discs in planar open books

Using open book foliations we show that an overtwisted disc in a planar open book can be put in a topologically nice position. As a corollary, we prove that a planar open book whose fractional Dehn twist coefficients grater than one for all the boundary components supports a tight contact structure.

Operations on open book foliations

We study b ‐arc foliation changes and exchange moves of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda’s bypass attachment operation. As applications, we study how open book foliations change under a stabilization of the open book. We also generalize Birman‐Menasco’s split/composite braid theorem: we show that closed braid representatives of a split (resp. composite) link in a certain open book can be converted to a split (resp. composite) closed braid by applying exchange moves finitely many times. 57M27

The self-linking number in annulus and pants open book decompositions

We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequins self-linking formula for a braid in the standard contact

Connect sum and transversely non simple knots

3

Khovanov-Rozansky homology and the braid index of a knot

-sphere.

POLYNOMIAL INVARIANTS OF PSEUDO-ANOSOV MAPS

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Veŕtesis result [ 11 ].