#### Date of Award

1979

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Graduate Group

Mathematics

#### First Advisor

Peter J. Freyd

#### Abstract

The usual algebraic construction used to study the symmetries of an object is the group of automorphisms of that object. In many geometric settings, however, one may interpret the symmetries in a more intimate manner by an algebraic structure on the object itself. Define a *quandle* to be a set equipped with two binary operations, (*x*,*y*) ↦ *x *▷ *y* and (*x*, *y*) ↦ *x* ▷^{-1}* y*, which satisfies the axioms

**Q1. ** *x* ▷ *x* = *x*.

**Q2. **(*x* ▷ *y*) ▷^{-1} *y* = *x* = (*x* ▷^{-1} *y*) ▷ *y.*

**Q3. ** (*x* ▷ *y*) ▷ *z* = (*x* ▷ *z*) ▷ (*y* ▷ *z*).

Call the map *S*(*y*) sending x to *x* ▷ *y* the *symmetry* at *y*.

To each point *y* of a symmetric space there is a symmetry *S*(*y*), the symmetric space becomes a quandle. Call a quandle satisfying *x* ▷ *y* = *x* ▷^{-1}* y* an *involutory* quandle. Loos [1] has defined a symmetric space as a manifold with an involutory quandle structure such that each point *y* is an isolated fixed point of *S*(*y*).

The underlying set of a group *G* along with the operations of conjugation, *x* ▷ *y* = *y*^{-1}*xy* and *x* ▷^{-1} *y* = *yxy*^{-1} form a quandle Conj *G*. Moreover, the theory of conjugation may be regarded as the theory of quandles in the sense that any equation in ▷ and ▷^{-1} holding in Conj *G* for all the groups *G* also holds in any quandle. If the center of *G* is trivial, then Conj *G* determines *G*.

Let *G* be a group and *n* ≥ 2. The *n-core* of *G* is the set

{(*x*_{1},*x*_{2},...,*x _{n}*) ∈

*G*|

^{n}*x*

_{1}

*x*

_{2}...

*x*= 1}

_{n}along with the operation

(*x*_{1},*x*_{2},...,*x _{n}*) ▷(

*y*

_{1},

*y*

_{2},...,

*y*) = (

_{n}*y*

_{n}^{-1}

*x*

_{n}y_{1}

^{-1}

*x*M

_{1}

*y*

_{2},...,

*y*

_{n-1}

^{-1}

*x*

_{n-1}

*y*).

_{n}The *n*-core is an *n-quandle*, that is, each symmetry has order dividing *n*. The group *G* is simple if and only if its *n*-core is a simple quandle.

Let *G* be a noncyclic simple group and *Q* a nontrivial conjugacy class in *H* viewed as a subquandle of Conj *G*. Then *Q* is a simple quandle.

Let *Q* be a quandle. The *transvection group* of *Q*, Trans*Q*, is the automorphism group of *Q* generated by automorphisms of the form *S*(*x*)*S*(*y*)^{-1} for *x, y* in *Q*. Suppose *Q* is a simple *p*-quandle where *p* is prime. Then either Trans*Q* is a simple group, or else *Q* is the *p*-core of a simple group *G* and Trans*Q* = *G ^{p}.*

Consider the category of pairs of topological spaces (*X,K*),* **K* ⊆ *X*, where a map *f *: (*X,K*) → (*Y,L*) is a continuous map *f* : *X* → *Y* such that *f*^{-1}(*L*) = *K*. Let (*D,O*) be the closed unit disk paired with the origin *O*. Call a map from (*D,O*) to (*X,K*) a *noose* in *X* about *K*. The homotopy classes of nooses in *X* about *K* form the *fundamental quandle Q*(*X,K*). The inclusion of the unit circle to the boundary of *D* gives a natural transfromation from *Q*(*X,K*) to the fundamental group *π*_{1}(*X - K*). A statement analogous to the Seifert-Van Kampen theorem for the fundamental group holds for the fundamental quandle.

Let *K* be an oriented knot in the 3-sphere *X*. Define the *knot quandle Q*(*K*) to be the subquandle of *Q*(*X,K*) consisting of nooses linking once with *K*. Then *Q*(*K*) is a classifying invariant of tame knots, that is, if *Q*(*K*) = *Q*(*K*′), then *K* is equivalent to *K*′. The knot group and the Alexander invariant can be computed from *Q*(*K*).

[1] Loos, O., *Symmetric Spaces*, Benjamin, New York, 1969.

#### Recommended Citation

Joyce, David Edward, "An Algebraic Approach to Symmetry With Applications to Knot Theory" (1979). *Publicly Accessible Penn Dissertations*. 952.

https://repository.upenn.edu/edissertations/952

## Comments

This edition of my dissertation differs little from the original 1979 version. This edition is typeset in LATEX, whereas the original was typed on typewriter and the figures were hand drawn. The page numbers and figure numbers are changed, the table of contents is expanded to include sections, a list of figures is included, and the index appears at the end instead of the front. I've corrected a few typos (and probably added others), and I added figure 4.5 that was missing from the original.

David Joyce

June, 2009