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
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
, and L(19,7)