## A (Z⊕Z)-family of knot quandles (Jim Hoste, Pitzer College)

### April 25 @ 12:00 pm - 1:30 pm

Suppose K is an oriented knot in a 3-manifold M with regular neighborhood N (K). For each element γ ∈ π 1 (∂N (K)) we define a quandle Q γ (K; M) which generalizes the concept of the fundamental quandle of a knot. In particular, when γ is the meridian of K, we obtain the fundamental quandle. The collection of all such quandles gives a (Z⊕Z)-family of quandles. If K is a knot in M and γ is a primitive element, then we show that there exists a knot K’ in a 3-manifold M’ such that Q γ (K; M ) ∼= Q μ (K’ ; M’) where μ is the meridian of K’ . Starting with a partially framed link L in the 3-sphere where the framed components give a surgery description of the manifold M and a single unframed component represents K we can derive a similar surgery description of K’ in M’ . Using results of Fenn and Rourke, we may then use this description of K’ to record a presentation of the quandle Q γ (K; M). We describe a number of examples of these quandles for knots

in various manifolds.