We define a group-valued invariant of virtual knots and relate it to various other group-valued invariants of virtual knots, including the extended group of Silver-Williams and the quandle group of Manturov and Bardakov-Bellingeri. A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that the group factors as a free product of the usual knot group and Z. We establish a similar formula for mod p almost classical knots, and we use these results to derive obstructions to a virtual knot K being mod p almost classical. Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to 6 crossings and determine their Alexander polynomials and virtual genus.Comment: 44 page

An accelerating liquid drop, under the action of surface tension, is shown to be unstable to small disturbances above a first critical value of the Bond number. Both numerical and second-order asymptotic methods are employed in order to characterize the normal-mode response and the neutral-stable modes at larger values of the Bond number. The transient response of an initially spherical drop that is accelerated by the flow of an external gas is studied as an initial-value problem. A unified theory, that includes acceleration as well as aerodynamic effects, is presented in order to account for the complete dynamic range of Weber and Bond numbers. The results are compared with experimental observations that range from continuous vibration to irreversible aerodynamic distortion and unstable shattering.

Abstract. Given a virtual knot K, we introduce a new group-valued invariant VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the 0-th ideal is a polynomial HK (s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial GK (s, t) introduced in [40,24,42]. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation : VGK → GLn(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

In 1992, Xiao-Song Lin constructed an invariant h(K ) of knots K ⊂ S 3 via a signed count of conjugacy classes of irreducible SU(2) representations of π 1 (S 3 − K ) with trace-free meridians. Lin showed that h(K ) equals one half times the knot signature of K . Using methods similar to Lin's, we construct an invariant h(L) of two-component links L ⊂ S 3 . Our invariant is a signed count of conjugacy classes of projective SU(2) representations of π 1 (S 3 − L) with a fixed 2-cocycle and corresponding nontrivial w 2 . We show that h(L) is, up to a sign, the linking number of L.

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer-Mrowka instanton Floer homology for links.