Coloring Link Diagrams And Conway-type Polynomial Of Braids

In this paper we define and present a simple combinatorial formula for a 3-variable Laurent polynomial invariant of conjugacy classes in Artin braid group $B_m$. We show that this Laurent polynomial satisfies the Conway skein relation and its coefficients are Vassiliev invariants of braids.Comment: This is a revised version. To appear in Topology and its Application

Link invariants via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a knot. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting (with signs and multiplicities) subdiagrams of certain combinatorial types. These formulas generalize the calculation of a linking number by counting signs of crossings in a link diagram. Until recently, explicit formulas of this type were known only for few invariants of low degrees. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. We then identify the resulting invariants with certain derivatives of the HOMFLYPT polynomial.Comment: This is a revised version, to appear in Geom. Dedicata, 29 pages, many figure

On the autonomous metric on groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces

Let $\Sigma_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(\Sigma_g)$ be the group of Hamiltonian diffeomorphisms of $\Sigma_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $Ham(\Sigma_g)$ is unbounded with respect to this metric.Comment: Now it is a part of arXiv:1405.793

Invariants of closed braids via counting surfaces

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.Comment: This is a revised version. To appear in JKTR. arXiv admin note: text overlap with arXiv:1209.042

The autonomous norm on Ham(R2n) is bounded

We thank the Center for Advanced Studies in Mathematics at Ben Gurion University for supporting the visit of the second author at BGU. We also thank the anonymous referee for useful comments.Peer reviewedPostprin

Concordance group and stable commutator length in braid groups

We define a quasihomomorphism from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, we provide a relation between the stable four ball genus in the concordance group and the stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. We also provide applications to the geometry of the infinite braid group. In particular, we show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich.Comment: 25 pages, 5 figure

On the autonomous metric on the groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces

Let g be a closed hyperbolic surface of genus g and let Ham g be the group of Hamiltonian diffeomorphisms of g. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that Ham g is unbounded with respect to this metri