Words and Quadratic Numbers

This chapter offers an overview of words and quadratic numbers, and in particular ordering the conjugates of a Christoffel word. Within this topic the reader learns that the reversal of a Christoffel word is a conjugate, and that the lower and upper Christoffel words of the same slope are the smallest and the largest in their conjugation class. The chapter discusses computation in terms ofMarkoff numbers of the quadratic real number which has a periodic continued fraction with periodic pattern equal to a Christoffel word written on the alphabet 11, 22. It also reviews computation of theMarkoff supremum of a periodic biinfinite sequence, and of theLagrange number of a periodic sequence, both having a periodic pattern as above.

Root allomorphy and conjugation class

This chapter considers the (limited) extent to which conjugation classes in the Romance verb interact with the morphomic patterns of root allomorphy discussed elsewhere. It is shown how, in Ibero-Romance, inflexion-class assignment shows a surprising sensitivity to the morphomic L-pattern. The general resistance (particularly with respect to palatalization) of first-conjugation verbs to morphomic patterns of root allomorphy in most Romance languages is also explored.

Periphrasis in the Sanskrit Verb System*

Ancient Sanskrit had two tenses of particular interest: periphrastic perfect and periphrastic future. At first glance, they are rather similar: both realize a particular value of tense through a combination of a lexical verb (devoid of personal agreement) and an agreeing auxiliary. There are, however, important differences which are revealed in this chapter: the periphrastic future is available for every verb, and can be distinguished from the synthetic future on semantic grounds, while the periphrastic perfect is available only for certain verbs, and these do not make up a semantically homogeneous group. A formal analysis is proposed, within Paradigm Function Morphology, for the two periphrastic tenses. It is demonstrated that a morphological rather than a purely syntactic account is preferable here. The verbs with a periphrastic perfect make up a conjugation class; on the other hand, the periphrastic future is formalized as a morphosyntactic property whose default realization is periphrastic.

WOVEN BRAIDS AND THEIR CLOSURES

A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence tame links can be presented as plats and closures of woven braids. Restricting on knots we get the 'woven version' of the well-known theorem of Markov, giving moves that are capable of producing all woven braids with equivalent closures. As corollary we obtain that a link in which each component is dyed with at least two different colors can be projected on a plane without crossing strands of the same color. The lowest order part of the HOMFLY polynomial of a closed woven braid can be read off directly; a complete characterization of all occuring terms is given. Finally, a table of all minimal woven braids for the 84 prime knots with at most nine crossings is appended. The average word length and the average number of entries per knot type tune out to be suprising small.

Lyndon Words, Free Algebras and Shuffles

A Lyndon word is a primitive word which is minimum in its conjugation class, for the lexicographical ordering. These words have been introduced by Lyndon in order to find bases of the quotients of the lower central series of a free group or, equivalently, bases of the free Lie algebra [2], [7]. They have also many combinatorial properties, with applications to semigroups, pi-rings and pattern-matching, see [1], [10].We study here the Poincaré-Birkhoff-Witt basis constructed on the Lyndon basis (PBWL basis). We give an algorithm to write each word in this basis: it reads the word from right to left, and the first encountered inversion is either bracketted, or straightened, and this process is iterated: the point is to show that each bracketting is a standard one: this we show by introducing a loop invariant (property (S)) of the algorithm. This algorithm has some analogy with the collecting process of P. Hall [5], but was never described for the Lyndon basis, as far we know.