The Turaev genus of an adequate knot

The Turaev genus of a knot is an obstruction to the knot being alternating. An adequate knot is a generalization of an alternating knot. A natural problem is a characterization of the Turaev genus of an adequate knot. In this paper, we show that the Turaev genus of an adequate knot is realized by the genus of the Turaev surface associated to an adequate diagram of the knot using the Khovanov homology. As a result, we obtain the additivity of the Turaev genus of adequate knots, and show that the Turaev genus of an adequate knot is “often” preserved under mutation. We also show that an n-semi-alternating knot is of Turaev genus n. This is the first examples of adequate knots of Turaev genus two or more.     

Local extrema in genus-stratified graphs

Beyond the obvious organization of all the orientable imbeddings of a graph according to the genus of the imbedding surface, there are several seemingly natural ways to ascribe proximity of imbeddings. One of these is to stipulate that two imbeddings are adjacent if one imbedding can be obtained from the other by moving one end of an edge in its rotation. If one associates “altitude” with genus, then one might hope to construct algorithms for minimum genus and maximum genus by descent and ascent, respectively. This investigation of the structure of the system of orientable graph imbeddings reveals that although there may occur arbitrarily deep traps among the local minima, there cannot exist any strict local maxima. These new discoveries seem consistent with a known contrast in the computational complexity of the maximum genus and minimum genus problems. That is, whereas Furst, Gross, and McGeoch have devised a polynomial-time algorithm to find the maximum genus of an arbitrary graph, Thomassen has proved that the problem of finding the minimum genus is NP-complete.     

Genera of Cayley maps

The genus distribution of a graph G is defined to be the sequence {gm}, where gm is the number of different embeddings of G in the closed orientable surface of genus m. In this paper, we examine the genus distributions of Cayley maps for several Cayley graphs. It will be shown that the genus distribution of Cayley maps has many different properties from its usual genus distribution.     

On the genus distributions of wheels and of related graphs

We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl’s result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels.     

On the average genus of a graph

Not all rational numbers are possibilities for the average genus of an individual graph. The smallest such numbers are determined, and varied examples are constructed to demonstrate that a single value of average genus can be shared by arbitrarily many different graphs. It is proved that the number 1 is a limit point of the set of possible values for average genus and that the complete graph K₄ is the only 3-connected graph whose average genus is less than 1.     

Numerical computation of the genus of an irreducible curve within an algebraic set

The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one-dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. The most important invariants of a curve are the degree, the arithmetic genus and the geometric genus (where the geometric genus denotes the genus of a desingularization of the projective closure of the curve). This article presents a numerical algorithm to compute the geometric genus of any one-dimensional irreducible component of an algebraic set.     

The sectional genus of quasi-polarised varieties

T. Fujita conjectured that the sectional genus of a quasi-polarised variety is non-negative. We prove this conjecture. Using the minimal model program we also prove that if the sectional genus is zero the Δ-genus is also zero. This leads to a birational classification of quasi-polarised varieties with sectional genus zero.     

Genus fields of Kummer extensions of rational function fields

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields. Our main result, the genus of a general Kummer extension of a global rational function field, is a direct consequence of this fact.     

On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus

We obtain an upper bound for the number of holomorphic mappings of a genus 3 Riemann surface onto a genus 2 Riemann surface in a series of cases. In particular, we establish that the number of holomorphic mappings of an arbitrary genus 3 Riemann surface onto an arbitrary genus 2 Riemann surface is at most 48. We show that this estimate is sharp and find pairs of Riemann surfaces for which it is attained.     

A refined Hurwitz theorem for imbeddings of irredundant Cayley graphs

Hurwitz's theorem states that the order of any finite group acting on a surface of genus γ > 1 is bounded by 168(γ ? 1). It can be refined to give useful information about groups whose order is near this bound. In this paper, similar results are obtained for Cayley graphs imbedded in a surface of genus γ. These results have important implications for the classification of Cayley graphs of low genus and the number of Cayley graphs of a given genus.     

The ordinarization transform of a numerical semigroup and semigroups with a large number of intervals

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (the Frobenius number). This gives another numerical semigroup and by repeating this transform several times we end up with an ordinary semigroup. The genus, that is, the number of gaps, is kept constant in all the transforms.This procedure allows the construction of a tree for each given genus containing all semigroups of that genus and rooted in the unique ordinary semigroup of that genus. We study here the regularity of these trees and the number of semigroups at each depth. For some depths it is proved that the number of semigroups increases with the genus and it is conjectured that this happens at all given depths. This may give some light to a former conjecture saying that the number of semigroups of a given genus increases with the genus.We finally give an identification between semigroups at a given depth in the ordinarization tree and semigroups with a given (large) number of gap intervals and we give an explicit characterization of those semigroups.     

Lower bounds for the average genus

Two lower bounds are obtained for the average genus of graphs. The average genus for a graph of maximum valence at most 3 is at least half its maximum genus, and the average genus for a 2-connected simplicial graph other than a cycle is at least 1/16 of its cycle rank. © 1995 John Wiley & Sons, Inc.     

There is one group of genus two

The genus of a finite group is the minimum genus over all surfaces containing an imbedded Cayley graph for the group. It is shown that there is exactly one group of genus two.     

Order,genus, and diagram invariance

The genus of any ordered set equals the genus of its covering graph, and, therefore, the genus of an ordered set is a diagram invariant.     

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.     

Function field genus theory for non-Kummer extensions

In this paper we first obtain the genus field of a finite abelian non-Kummer l–extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field with relatively prime degrees is equal to the composite of their respective genus fields and our previous results, we deduce the general expression of the genus field of a finite abelian extension of a global rational function field.     

On the minimal genus of 2-complexes

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking ⌊ log₂ g⌋ th barycentric subdivision is not sufficient, where g is the genus of K. On the other hand, (1+⌈log₂(g+2)⌉)th subdivision is proved to be sufficient. © 1997 John Wiley & Sons, Inc.     

Hypermaps versus bipartite maps

A hypermap was defined by R. Cori to be a pair of permutations σ and α on a finite set B, such that the group generated by σ and α is transitive on B. The genus of a hypermap was defined according to a formula of A. Jacques for the genus of a pair of permutations. This paper presents a one-to-one correspondence between the set of hypermaps of a given genus and the set of 2-colored bipartite maps of the same genus.     

A relative maximum genus graph embedding and its local maximum genus

A relative embedding of a connected graph is an embedding of the graph in some surface with respect to some closed walks, each of which bounds a face of the embedding. The relative maximum genus of a connected graph is the maximum of integerk with the property that the graph has a relative embedding in the orientable surface withk handles. A polynomial algorithm is provided for constructing relative maximum genus embedding of a graph if the relative tree of the graph is planar. Under this condition, just like maximum genus embedding, a graph does not have any locally strict maximum genus.     

Embeddings of cartesian products of nearly bipartite graphs

Surgical techniques are often effective in constructing genus embeddings of cartesian products of bipartite graphs. In this paper we present a general construction that is “close” to a genus embedding for cartesian products, where each factor is “close” to being bipartite. In specializing this to repeated cartesian products of odd cycles, we are able to obtain asymptotic results in connection with the genus parameter for finite abelian groups.