Narrow Results

Ito–Takimura recently defined a splice-unknotting number $u^{-}(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same result.) As an application, we compute the crosscap numbers of all prime alternating knots through at least 13 crossings, using Gauss codes.

This paper begins the development of a theory of what we will call symplectic alternating algebras. They have arisen in the study of 2-Engel groups but seem also to be of interest in their own right. The main part of the paper deals with the challenging classification of some algebras of this kind which arise in the context of 2-Engel groups and give some new information about these groups. The main result is that there are 31 such algebras with dimension 6 over the field of three elements.

In this paper and its sequel we continue our study of nilpotent symplectic alternating algebras. In particular we give a full classification of such algebras of dimension 10 over any field. It is known that symplectic alternating algebras over GF(3) correspond to a special rich class ${𝒞}$ of 2-Engel 3-groups of exponent 27 and under this correspondence we will see that the nilpotent algebras correspond to a subclass of ${𝒞}$ that are those groups in ${𝒞}$ that have an extra group theoretical property that we refer to as being powerfully nilpotent and can be described also in the context of $p$-groups where $p$ is an arbitrary prime.

In this paper, we finish our classification of nilpotent symplectic alternating algebras of dimension 10 over any field $𝔽$.

We continue developing the theory of nilpotent symplectic alternating algebras. The algebras of upper bound nilpotent class, that we call maximal algebras, have been introduced and well studied. In this paper, we continue with the external case problem of algebras of minimal nilpotent class. We show the existence of a subclass of algebras over a field $F$ that is of certain lower bound class that depends on the dimension only. This suggests a minimal bound for the class of nilpotent algebras of dimension $2n\ge 8$ of rank $2$ over any field.

In this paper, we will explore the properties of knots that lie in a surface cross an interval. We show that any alternating knot nontrivially embedded in S × I is chiral. This demonstrates that knots in S × I behave differently from knots in the three sphere, where alternating knots (like the figure eight) can be amphichiral. We use a generalized Kauffman bracket polynomial, as well as geometric and covering space techniques.

It is well-known that the "all right" metric structure on the Dehn complex of a knot is non-positively curved, or CAT(0), if and only if the knot projection is prime and alternating. We extend these ideas to spatial graphs and introduce the idea of ψ-alternating graphs as the appropriate analog of alternating knots.

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram $D$, it is known that $s(D)$, the number of Seifert circles in $D$, equals the braid index $b(D)$ of $D$ if $D$ contains no lone crossings, where a crossing in $D$ is called a lone crossing if it is the only crossing between two Seifert circles in $D$. If $D$ contains lone crossings, then one can reduce the number of Seifert circles in $D$ using link-type preserving moves such as the Murasugi–Przytycki operation. Let $r(D)\ge 1$ be the maximum number of Seifert circles in $D$ that can be reduced, then we have $b(D)\le s(D)-r(D)$. On the other hand, if the $a$-span of the HOMFLY polynomial of $D$ satisfies the equality $a$-span$/2+1=s(D)-r(D)$, then the Morton–Frank–Williams (MFW) inequality $a$-span$/2+1\le b(D)$ leads us to the simple braid index formula $b(D)=s(D)-r(D)$. In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality $a$-span$/2+1=s(D)-r(D)$. Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.

We tabulate all prime alternating knots through fourteen crossings. This provides the first list of 14-crossing prime alternating knots as well as the first confirmation of Thistlethwaite's tabulation of 12- and 13-crossing alternating knots.