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For an oriented link L C S3 = dD4 , let Xs(L) be the greatest Euler characteristic x(F) of an oriented 2-manifold F (without closed components) smoothly embedded in D4 with boundary L . A knot K is slice if Xs(K) = 1 . Realize D4 in C2 as {(z, w) : \z\2 + \w\2 < 1} . It has been conjectured that, if V is a nonsingular complex plane curve transverse to S3 ,(More)
Let S(D) be the surface produced by applying Seifert’s algorithm to the oriented link diagram D. I prove that if D has no negative crossings then S(D) is a quasipositive Seifert surface, that is, S(D) embeds incompressibly on a fiber surface plumbed from positive Hopf annuli. This result, combined with the truth of the “local Thom Conjecture”, has various(More)
LET f (z,w) ≡ f0(z)w + f1(z)w + · · · + fn(z) ∈ C[z,w]. Classically, the equation f (z,w) = 0 was said to define w as an (n-valued) algebraic function of z, provided that f0(z) was not identically 0 and that f (z,w) was squarefree and without factors of the form z− c. Then, indeed, the singular set B = {z: there are not n distinct solutions w to f (z,w) =(More)
The primary objects of study in the “knot theory of complex plane curves” are C-links: links (or knots) cut out of a 3-sphere in C2 by complex plane curves. There are two very different classes of C-links, transverse and totally tangential. Transverse C-links are naturally oriented. There are many natural classes of examples: links of singularities; links(More)
Given a linkage belonging to any of several broad classes (both planar and spatial), we have deflned parameters adapted to a stratiflcation of its deformation space (the quotient space of its conflguration space by the group of rigid motions) making that space \practically piecewise convex". This leads to great simpliflcations in motion planning for the(More)
How can a complex curve be placed in a complex surface? The question is vague; many different ways to make it more specific may be imagined. The theory of deformations of complex structure, and their associated moduli spaces, is one way. Differential geometry and function theory, curvatures and currents, could be brought in. Even the generalized Nevanlinna(More)
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon. Trivially, the connected sum (i.e., 2-gonal Murasugi sum) of ribbon knots is ribbon. Non-trivially [6, 1], any Murasugi sum of fibered knots(More)
Inverse kinematics (IK) problems are important in the study of robotics and have found applications in other fields such as structural biology. The conventional formulation of IK in terms of joint parameters amounts to solving a system of nonlinear equations, which is considered to be very hard for general chains, especially for those with many links. In(More)
Conventionally, joint angles are used as parameters for a spatial chain with spherical joints, where they serve very well for the study of forward kinematics (FK). However, the inverse kinematics (IK) problem is very difficult to solve directly using these angular parameters, on which complex nonlinear loop closure constraints are imposed by required end(More)