Free quandles and knot quandles are residually finite

Abstract

In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual finiteness of automorphism groups of some residually finite quandles is also discussed.2010 Mathematics Subject Classification. Primary 57M25; Secondary 20E26, 57M05, 20N05. Key words and phrases. 3-manifold group, enveloping group, free quandle, knot group, knot quandle, peripheral subgroup, residually finite quandle.

Multi-switches and virtual knot invariants

“…Over the years, quandles have been investigated by various authors for constructing new invariants for knots and links (see, for example, [15,17,25,29,34,36]). Algebraic properties of quandles including their automorphisms and residual properties have been investigated, for example, in [1,6,9,12,18,37]. For more details about quandles see [15,22,38].…”

“…and using the second axiom of biquandles we can exlude x n+1 from equalities (8), (9) and obtain the equality xi = xj . Therefore X S,V (D 1 ) is the quotient of X (n) by the relations which can be written from the part of the diagram outside of Figure 9 and m + 1 relations (10) xi = xj .…”

Subquandles of free quandles

“…As for any type of algebraic structure, in quandle theory there is a notion of a free quandle. V. Bardakov, M. Singh and M. Singh in [1,Problem 6.12] raised the question about an analogue of Nielsen-Schreier theorem for quandles: is it true that any subquandle of a free quandle is free. This note is devoted to an affirmative answer on this question:…”

General constructions of biquandles and their symmetries

“…In order to obtain reasonably strong knot invariants from quandles, it is necessary to understand them from algebraic point of view. Algebraic properties of quandles including their automorphisms and residual properties have been investigated, for example, in [2,3,7,8,16,32,33,46,48]. A (co)homology theory for quandles and racks has been developed in [13,25,26,47], which, as applications has led to stronger invariants for knots and links.…”