• A small and a large Seifert surface of a minimal Morse map u for U, the unlink of two components.
• Level sets of height on a standard 1-patch and a standard 2-patch.
• Piecewise-smooth versions of the standard 1-patch and the standard 2-patch.
• Two hats.
• An o-good neighborhood of an n-star on a disk bounded by the unknot O (fibered by o), n=1, 2, 3.
• A ``side view'' (indicating some level sets of a Morse map f) of the part of a non-f-good neighborhood M near an endpoint of an n-star.
• Plumbing and twisting.
• Piecewise-smooth versions of a small and a large Seifert surface of u.
• Piecewise-smooth Seifert surfaces (a disk, and a punctured torus with viewport) for a plumbing of u to itself.
• Hopf plumbing undoes twisting.
• A proper arc on the fiber surface of the positive trefoil knot; the fiber surface with an unknotted, untwisted annulus plumbed to it along that arc.
• A non-free knotted annulus obtained by cutting the fiber surface of the positive trefoil knot along a proper arc; a non-free punctured torus which is a large Seifert surface of a minimal Morse map for a twisted double of the trefoil.
• Unplumbing an inhomogeneous braided surface of degree 2.

• A 3-string cable as a 6-gonal Murasugi sum.

• Three Legendrian knots with their framings.
• A (-1)-twisted +-clasped chain of 4 unknots of type O.
• The Legendrian knots above, turned into chains which are links at infinity.
• A divide which is an arrangement, and its A'Campo link as a link at infinity.
• Distinct real arrangments with identical complexifications.

• A degree-4 polynomial p as a simple covering of C by C.
• The inverse of p as a map from the regular values of p to the configuration space of unordered 4-tuples in C.
• Two views of a homogeneous pre-braid.

• Top-plumbing an annulus to a Seifert surface.
• Bottom-plumbing an annulus to a Seifert surface.
• Indifference of Hopf annuli to top and bottom.
• Boundary-connected sum as plumbing.
• Any Murasugi sum with an annulus is an annular plumbing.
• A case when top- and bottom-plumbing are the same.
• The boundaries of top- and bottom-plumbings are always the same.
• Sliding one end of a knotted 1-handle over an unknotted 1-handle.
• Pushing the transverse arc off the 1-handle.
• Elementary moves of fence diagrams.
• Fence diagrams and espaliers.
• More fence diagrams and espaliers.
• Boundaries of compression disks on two inhomogeneous bandword surfaces.

• The three Reidemeister moves.
• Cover art for the January, 1995, issue of the American Scientist.

• A braided Seifert surface and a corresponding charged fence diagram.
• A deflation; two slips.
• Two slides up; two slides down.
• A twirl; a turn.
• A bent slide; a doubly-bent slide.
• Three steps in the proof that every Stallings plumbing is realizable as a braided Stallings plumbing.
• Three more steps in the proof that every Stallings plumbing is realizable as a braided Stallings plumbing.
• The last three steps in the proof that every Stallings plumbing is realizable as a braided Stallings plumbing.

[Last updated April 24, 2001]