Narrow Results

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

Applying the concept of braiding sequences and the inequality between the signature and number of roots of the Alexander polynomial on the unit circle, we prove that only finitely many special alternating knots are (even algebraically) concordant, in that their concordance class determines their Alexander polynomial. We discuss some extensions of this result to positive and almost positive knots, and links.

An unfolding of a polyhedron is a single connected planar piece without overlap resulting from cutting and flattening the surface of the polyhedron. Even for orthogonal polyhedra, it is known that edge-unfolding, i.e., cuts are performed only along the edges of a polyhedron, is not sufficient to guarantee a successful unfolding in general. However, if additional cuts parallel to polyhedron edges are allowed, it has been shown that every orthogonal polyhedron of genus zero admits a grid-unfolding with quadratic refinement. Using a new unfolding technique developed in this paper, we improve upon the previous result by showing that linear refinement suffices. For 1-layer orthogonal polyhedra of genus $g$, we show a grid-unfolding algorithm using only $2(g-1)$ additional cuts, affirmatively answering an open problem raised in a recent literature. Our approach not only requires fewer cuts but yields much simpler algorithms.

Given two numerical semigroups S and T and a positive integer d, S is said to be one over d of T if S = {s ∈ ℕ | ds ∈ T} and in this case T is called a d-fold of S. We prove that the minimal genus of the d-folds of S is , where g and f denote the genus and the Frobenius number of S. The case d = 2 is a problem proposed by Robles-Pérez, Rosales, and Vasco. Furthermore, we find the minimal genus of the symmetric doubles of S and study the particular case when S is almost symmetric. Finally, we study the Frobenius number of the quotient of some families of numerical semigroups.

In this work, we give parametrizations in terms of the Kunz coordinates of numerical semigroups with multiplicity up to $5$. We also obtain parametrizations of MED semigroups, symmetric and pseudo-symmetric numerical semigroups with multiplicity up to $5$. These parametrizations also lead to formulas for the number of numerical semigroups, the number of MED semigroups and the number of symmetric and pseudo-symmetric numerical semigroups with multiplicity up to $5$ and given conductor.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number $F$, genus $g$ and type $t$. It is known that for any numerical semigroup $\frac{g}{F+1-g}\le t\le 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $\alpha$ the number of numerical semigroups with Frobenius number $F$ and type $F-\alpha$ is eventually constant for large $F$. The number of numerical semigroups with genus $g$ and type $g-\alpha$ is also eventually constant for large $g$.

In this work, we study $A$-semigroups, that is, numerical semigroups which have no consecutive elements less than the Frobenius number. We give algorithms that allow computation of the whole set of $A$-semigroups with a given genus, multiplicity and Frobenius number and from this we study interesting families of $A$-semigroups which are Frobenius varieties, pseudo-varieties; R-varieties.

In this paper, we show that g_c(K_{m, n}) - g(K_{m, n}) = 2mn for m, n ∈ N, where K_{m, n} is the connected sum of m copies of any Whitehead double of the (2n + 1, 2) torus knot.

This paper is an unpublished version of a talk that the author gave in Brazil in 1987 at a meeting of the Brazilian Mathematical Society, while visiting Sostenes Lins in Recife, Brazil. The paper is an introduction to state models for link invariants, and it gives a surface-genus interpretation of the extreme terms in the bracket polynomial.

For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K) : c(K) ≤ ⌊(g(K)+9)/6⌋ and c(K) ≤ ⌊(n(K) + 16)/12⌋. The (6n - 2,3) torus knots show that these bounds are sharp.

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we analyze the trivializing number of a pseudodiagram, i.e. the minimum number of crossings that must be resolved to produce the unknot. We consider how much crossing information is needed in a virtual pseudodiagram to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

Every finite field 𝔽_q, q = pⁿ, carries several Alexander quandle structures 𝕏 = (𝔽_q, *). We denote by the family of these quandles, where p and n vary respectively among the odd primes and the positive integers.

For every k-component oriented link L, every partition of L into sublinks, and every labeling of such a partition, the number of 𝕏-colorings of any diagram of is a well-defined invariant of , of the form for some natural number . Letting 𝕏 and vary respectively in and among the labelings of , we define the derived invariant .

If is such that , we show that , where t(L) is the tunnel number of L, generalizing a result by Ishii. If is a "boundary partition" of L and denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the L_j's, then we show that . We point out further properties of , mostly in the case of , . By elaborating on a suitable version of a result by Inoue, we show that when L = K is a knot then , where is the breadth of the Alexander polynomial of K. However, for every g ≥ 1 we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants . Moreover, in such examples provides sharp lower bounds for the genera of the knots. On the other hand, we show that can give better lower bounds on the genus than , when L has k ≥ 2 components.

We show that in order to compute it is enough to consider only colorings with respect to the constant labeling . In the case when L = K is a knot, if either or provides a sharp lower bound for the knot genus, or if , then can be realized by means of the proper subfamily of quandles {𝕏 = (𝔽_p, *)}, where p varies among the odd primes.

In this paper, we associate a plane graph G with an oriented link by replacing each vertex of G with a special oriented n-tangle diagram. It is shown that such an oriented link has the minimum genus over all orientations of its unoriented version if its associated plane graph G is 2-connected. As a result, the genera of a large family of unoriented links are determined by an explicit formula in terms of their component numbers and the degree sum of their associated plane graphs.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is an embedding under which the formal adjacency relation on half-edges corresponds to the adjacency relation induced by the embedding. *-graphs are a natural generalization of four-valent framed graphs, which are four-valent graphs with an opposite half-edge structure. In [Embeddings of four-valent framed graphs into 2-surfaces, Dokl. Akad. Nauk424(3) (2009) 308–310], the question of whether a four-valent framed graph admits a ℤ₂-homologically trivial embedding into a given surface was shown to be equivalent to a problem on matrices. We show that a similar result holds for *-graphs in which all vertices have degree 4 or 6. This gives an algorithm in quadratic time to determine whether a *-graph admits an embedding into the plane.

Let u(K) and g(K) denote the unknotting number and the genus of a knot K, respectively. For a 3-braid knot K, we show that u(K) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image.

The concordance genus of a knot is the least genus of any knot in its concordance class. It is bounded above by the genus of the knot, and bounded below by the slice genus, two well-studied invariants. In this paper we consider the concordance genus of 11-crossing prime knots. This analysis resolves the concordance genus of 533 of the 552 prime 11-crossing knots. The appendix to the paper gives concordance diagrams for 59 knots found to be concordant to knots of lower genus, including null-concordances for the 30 11-crossing knots known to be slice.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper [Embeddings of *-graphs into 2-surfaces, preprint (2012), arXiv:1212.5646] by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admits a ℤ₂-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices. Here we extend those results to nonorientable surfaces. The embeddability condition that we obtain yields quadratic-time algorithms to determine whether a *-graph with all vertices of degree 4 or 6 admits a ℤ₂-homologically trivial embedding into the projective plane or into the Klein bottle.

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot K such that $a(K)=2$ and $g(K)=n$, and that there exists a knot K’ such that $a(K^{\prime })\ge n$ and $g(K^{\prime })=1$ for any positive integer n. We also give an upper bound of the ascending number for a 2-bridge knot.

We show that the differences between canonical genus and free genus, and differences between free genus and usual genus of a knot can be arbitrarily large.