The three cyclic surgeries
of the
(-2,3,7) Pretzel knot
The (-2,3,7) Pretzel knot K
may be obtained by Dehn surgery on the minimally twisted five chain
link, MT5C. K will be the core of one of the surgery solid
tori. It is also an embedded curve on the once-punctured torus fiber
of the trefoil knot. Viewing this fiber as the plumbing of two Hopf
bands, K may be obtained by Dehn twisting the first core twice
about the second and then Dehn twisting this result twice about the
original first core.
Quotienting each of these constructions by an involution, we may view the exterior of K as a ball containing a 2-string tangle as the fixed set of the involution. The presentation as a "punctured" 3-strand braid naturally arises from each of these constructions.
On the left, the colored arcs suggest the tangles filling four of the boundary components of the quotiented exterior of the MT5C. On the right, the colored arcs taken in pairs suggest the tangles descending from the Dehn surgery realization of the Dehn twists on the trefoil.
Below we plug the above tangle with three different rational tangles. Then we perform isotopies to obtain two-bridge presentations of the resulting links. In this manner we may observe how the exterior of K admits the three cyclic fillings S^3, L(18,5), and L(19,7).
Quotienting each of these constructions by an involution, we may view the exterior of K as a ball containing a 2-string tangle as the fixed set of the involution. The presentation as a "punctured" 3-strand braid naturally arises from each of these constructions.
On the left, the colored arcs suggest the tangles filling four of the boundary components of the quotiented exterior of the MT5C. On the right, the colored arcs taken in pairs suggest the tangles descending from the Dehn surgery realization of the Dehn twists on the trefoil.
Below we plug the above tangle with three different rational tangles. Then we perform isotopies to obtain two-bridge presentations of the resulting links. In this manner we may observe how the exterior of K admits the three cyclic fillings S^3, L(18,5), and L(19,7).