We describe an alternative way of computing Alexander polynomials of knots/links, based on the Artin representation of the corresponding braids by automorphisms of a free group. We apply the same method to other representations of braid groups discovered by Wada and compare the corresponding isotopic invariants to Alexander polynomials.
It is proved that (a) there exists a complete countable theory having 2ω countable models, some inessential extension of which has ω countable models, and that (b) there exists a complete countable theory having 2ω countable models, some inessential extension of which has finitely many countable models. This gives an answer to the question of A. D. Taimanov.
This short note contains necessary addenda and corrections to [1].
We introduce a new version of orthogonality for nonalgebraic 1-types in weakly o-minimal theories, that of almost quite orthogonality. It is stated that the non almost quite orthogonality relation is an equivalence relation. The main result is a criterion for a weakly o-minimal theory of finite convexity rank with 1 < I(T, ω) < 2ω to have exactly 3m6l countable pairwise nonisomorphic models for some nonnegative integers m, l < ω in terms of the orthogonality version presented.
We continue and finalize our research started in [Algebra and Logic, 63, No. 3 (2024), 168–178]. It is proved that there exists a noncomputable low c.e. set A such that any set that is CEA(A) and 2-c.e. has a c.e. degree.
We show that for a given (reduced) root system Σ and for any algebraically closed field k with zero characteristic, the validity of the Gelfand–Kirillov conjecture for the finite-dimensional Lie algebra ({mathfrak{g}}_{mathrm{k},Sigma },) the sole semisimple Lie algebra over k with root system Σ, is equivalent to the provability of a certain first-order sentence in the language of rings (mathcal{L})(0, 1, +, ∗, −) in the theory ACF0 of algebraically closed fields of zero characteristic.
A finitely generated group that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (GBS group). Every GBS group is the fundamental group π1(𝔸) of a suitable labeled graph 𝔸. If 𝔸 is a labeled tree, then we call π1(𝔸) a tree GBS group. It is known that GBS groups isomorphic to tree groups are themselves tree groups. It is shown that GBS groups universally equivalent to tree groups do not have to be tree groups. Moreover, we give a description of all GBS groups that are universally equivalent to tree groups.
We prove the theorem stating the following. Let G be a locally finite group saturated with groups from a set 𝔐 consisting of direct products of d dihedral groups. Then G is a direct product of d groups of the form B ⋋ <υ>, where B is a locally cyclic group inverted by an involution υ.
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