The problem of independence and completeness of rotation moment invariants is addressed in this paper. General method for constructing invariants of arbitrary orders by means of complex moments is described. It is shown that for any set of invariants there exists relatively small basis by means of which all other invariants can be generated. The method how to construct such a basis is presented...

In this paper, we give a complete set of finite type string link invariants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussar...

Contents 1 Introduction 2 2 Finiteness 13 3 The Conway polynomial 20 4 Finite type invariants from quantum invariants 30 5 Combinatorial structure of finite type invariants 47 6 Finite type invariants for spin manifolds 58 7 Finite type invariants for bounded manifolds 61 8 Finite type invariants for marked manifolds 62 9 Further generalizations 65 10 Relationships with other theories and other...

In this paper, we give a complete set of finite type string link invariants of degree < 5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussar...

We study the asymptotics of a family of link invariants on the orbits of a smooth volume-preserving ergodic vector field on a compact domain of the 3-space. These invariants, called linear saddle invariants, include many concordance invariants and generate an infinite-dimensional vector space of link invariants. In contrast, the vector space of asymptotic linear saddle invariants is 1-dimension...

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D. Ebin’s long-time existence result for geodesics on the symplectomorphism group to metrics not necessarily compatible with the symplectic structure. We also study the dynamics of symplectic point vortices, desc...

A method is presented for the stress analysis of flight vehicles under different flight conditions including gust and control surface deflection (or maneuver) using the governing equations of rigid-body motions and elastic deformations. The Lagrangian approach is used to derive the governing equations of motions. For this purpose, the basic equations of motions are derived in terms of potential...

The theory of knot invariants of finite type (Vassiliev invariants) is described. These invariants turn out to be at least as powerful as the Jones polynomial and its numerous generalizations coming from various quantum groups, and it is conjectured that these invariants are precisely as powerful as those polynomials. As invariants of finite type are much easier to define and manipulate than th...

For an n-component link, Milnor’s isotopy invariants are defined for each multi-index I = i1i2...im (ij ∈ {1, ..., n}). Here m is called the length. Let r(I) denote the maximum number of times that any index appears in I. It is known that Milnor invariants with r = 1, i.e., Milnor invariants for all multi-indices I with r(I) = 1, are link-homotopy invariant. N. Habegger and X. S. Lin showed tha...

We introduce an algebraic structure we call semiquandles whose axioms are derived from flat Reidemeister moves. Finite semiquandles have associated counting invariants and enhanced invariants defined for flat virtual knots and links. We also introduce singular semiquandles and virtual singular semiquandles which define invariants of flat singular virtual knots and links. As an application, we u...