It is proved that every knot in the major subfamilies of J. Berge’s lens space surgery (i.e., knots yielding a lens space by Dehn surgery) is presented by an L-shaped (real) plane curve as a divide knot defined by N. A’Campo in the context of singularity theory of complex curves. For each knot given by Berge’s parameters, the corresponding plane curve is constructed. The surgery coefficients ar...

Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in S3. We will prove this conjecture for null-homologous knots in arbitrary closed 3-manifolds. Namely, if K is a knot in a closed 3-manifold Y , Y −K is irreducible, and ̂ HFK(Y, K ) is monic, then K is fibred. The proof relies on previous works due to Gabai, Ozsváth–Szabó, Ghiggini and the author. A corollary is that i...

Vector cross product structures on manifolds include symplectic, volume, G2and Spin (7)-structures. We show that their knot spaces have natural symplectic structures, and we relate instantons and branes in these manifolds with holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi-Yau manifold de…nes a complex vector cro...

Conformally invariant functionals on the space of knots are introduced via extrinsic confor-mal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does,...

We give a self-contained treatment of Le and Habiro’s approach to the Jones function of a knot and Habiro’s cyclotomic form of the Ohtsuki invariant for manifolds obtained by surgery around a knot. On the way we reproduce a state sum formula of Garoufalidis and Le for the colored Jones function of a knot. As a corollary, we obtain bounds on the growth of coefficients in the Ohtsuki series for m...

Conformally invariant functionals on the space of knots are introduced via extrinsic confor-mal geometry of the knot and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does,...

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap number is smaller than or equal to the concordance crosscap number. We constr...

We show that every rational knot K of crossing number N admits a polynomial parametrization x = Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a = 3 and b + degC = 3N. We show that every rational knot also admits a polynomial parametrization with a = 4. If C(t) = Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic k...

PURPOSE The authors present and describe an original adaptation for the use of "boatman's knot" in renal vein ligation during laparoscopic nephrectomy. This procedure may replace the need for the endovascular stapler, which is considered the standard of care, but not available in several institutions in Brazil. The knot presented is also known as the "pig's knot" in several farms in Brazil. M...

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with th...