# ALEXANDER QUANDLES OF ORDER 16

@article{Murillo2004ALEXANDERQO, title={ALEXANDER QUANDLES OF ORDER 16}, author={Gabriel Murillo and Sam Nelson}, journal={Journal of Knot Theory and Its Ramifications}, year={2004}, volume={17}, pages={273-278} }

Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of distinct connected finite Alexander quandles.

## 7 Citations

On properties of commutative Alexander quandles

- Mathematics
- 2014

An Alexander quandle Mt is an abelian group M with a quandle operation a * b = ta + (1 - t)b where t is a group automorphism of the abelian group M. In this paper, we will study the commutativity of…

A SURVEY OF QUANDLE IDEAS

- Mathematics
- 2011

This article surveys many aspects of the theory of quandles which algebraically encode the Reidemeister moves. In addition to knot theory, quandles have found applications in other areas which are…

Distributivity in Quandles and Quasigroups

- Mathematics
- 2014

Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in…

REVISIT TO CONNECTED ALEXANDER QUANDLES OF SMALL ORDERS VIA FIXED POINT FREE AUTOMORPHISMS OF FINITE ABELIAN GROUPS

- Mathematics
- 2014

Abstract. In this paper we provide a rigorous proof for the fact thatthere are exactly 8 connected Alexander quandles of order 2 5 by com-bining properties of xed point free automorphisms of nite…

FINITE MODULES OVER ℤ[t, t-1]

- Mathematics
- 2012

Let Λ = ℤ[t, t-1] be the ring of Laurent polynomials over ℤ. We classify all Λ-modules M with |M| = pn, where p is a prime and n ≤ 4. Consequently, we have a classification of Alexander quandles of…

Finite Modules over $\Bbb Z[t,t^{-1}]$

- Mathematics
- 2011

Let $\Lambda=\Bbb Z[t,t^{-1}]$ be the ring of Laurent polynomials over $\Bbb Z$. We classify all $\Lambda$-modules $M$ with $|M|=p^n$, where $p$ is a primes and $n\le 4$. Consequently, we have a…

On Connected Component Decompositions of Quandles

- MathematicsTokyo Journal of Mathematics
- 2019

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots,…

## References

SHOWING 1-4 OF 4 REFERENCES

State-sum invariants of knotted curves and surfaces from quandle cohomology

- Mathematics
- 1999

State-sum invariants for classical knots and knotted surfaces in 4-space are developed via the cohomology theory of quandles. Cohomology groups of quandles are computed to evaluate the invariants.…

Classification of Finite Alexander Quandles

- Mathematics
- 2002

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander…

Figure 3: Results of computation of Im(Id − φ) for Alexander quandles given by automorphisms

- Figure 3: Results of computation of Im(Id − φ) for Alexander quandles given by automorphisms