We show that, for any fixed genus g, the ordinary generating function for the genus g partitions of an n-element set into k blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus 2 ...

We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3and 4-move conjectures, and the calculation of the maximal hyperbolic volume for weak genus two knots. We also study the values of the link polynomial...

Two new genera of crinoid-associated pontoniine shrimps, Brucecaris, new genus, and Unguicaris, new genus, are described and illustrated. Brucecaris is established for Periclimenes tenuis Bruce. Unguicaris panglaonis, new genus, new species, is described from crinoid-associated material collected at Panglao Island, Bohol Sea, The Philippines. Three species, Periclimenes alegrias Bruce, Periclim...

We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (Z/2Z) for any hyperelliptic genus 3 curve. These algorithms provide a rational isogeny...

This paper deals with computing topological invariants such as connected components, boundary surface genus, and homology groups. For each input data set, we have designed or implemented algorithms to calculate connected components, boundary surfaces and their genus, and homology groups. Due to the fact that genus calculation dominates the entire task for 3D object in 3D space, in this paper, w...

In his book [Ad, p. 105 bottom], C. Adams mentions a result of Morton that there exist knots, whose genus g is strictly less than their weak genus g̃, the minimal genus of (the surface of Seifert’s algorithm applied on) all their diagrams. This observation appears just as a remark in [Mo], but was very striking to the author. Motivated by Morton’s example, the author started in a series of paper...

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We express the genus-two fixed-complex-structure enumerative invariants of P and P in terms of the genus-zero enumerative invariants. The approach is to relate each genus-two fixedcomplex-structure enumerative invariant to the corresponding symplectic invariant.

We define the singular elliptic genus for arbitrary normal surfaces, prove that it is a birational invariant, and show that it generalizes the singular elliptic genus of Borisov and Libgober and the stringy χy genus of Batyrev and Veys.

We consider an important class of deformations of the genus zero bihamiltonian structure defined on the loop space of semisimple Frobenius manifolds, and present results on such deformations at the genus one and genus two approximations.