Suppose K is an oriented knot in a 3-manifold M with regular neighborhood N (K). For each element \u03b3 \u2208 \u03c0 1 (\u2202N (K)) we define a quandle Q \u03b3 (K; M) which generalizes the concept of the fundamental quandle of a knot. In particular, when \u03b3 is the meridian of K, we obtain the fundamental quandle. The collection of all such quandles gives a (Z\u2295Z)-family of quandles. If K is a knot in M and \u03b3 is a primitive element, then we show that there exists a knot K’ in a 3-manifold M’ such that Q \u03b3 (K; M ) \u223c= Q \u03bc (K’ ; M’) where \u03bc 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 \u03b3 (K; M). We describe a number of examples of these quandles for knots
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Suppose K is an oriented knot in a 3-manifold M with regular neighborhood N (K). For each element \u03b3 \u2208 \u03c0 1 (\u2202N (K)) we define a quandle Q \u03b3 […]<\/p>\n","protected":false},"author":105,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":""},"tags":[],"tribe_events_cat":[17],"class_list":["post-1289","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-topology-seminar","cat_topology-seminar"],"acf":[],"yoast_head":"\n