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For every odd integer $N$ we give an explicit construction of a polynomial curve $cC(t) = (x(t), y (t))$, where $deg x = 3$, $deg y = N + 1 + 2pent N4$ that has exactly $N$ crossing points $cC(t_i)= cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pade approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.
We show that no torus knot of type $(2,n)$, $n>3$ odd, can be obtained from a polynomial embedding $t mapsto (f(t), g(t), h(t))$ where $(deg(f),deg(g))leq (3,n+1) $. Eventually, we give explicit examples with minimal lexicographic degree.
Conway-normalized Alexander polynomial of ribbon knots depend only on their ribbon diagrams. Here ribbon diagram means a ribbon spanning the ribbon knot marked with the information of singularities. We further give an algorithm to calculate Alexander
We prove that the signature bound for the topological 4-genus of 3-strand torus knots is sharp, using McCoys twisting method. We also show that the bound is off by at most 1 for 4-strand and 6-strand torus knots, and improve the upper bound on the as
Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi-Yau geometry and refined Chern-Simons invariants of torus knots. Generalizing the untwisted case, the present appro
We recursively determine the homotopy type of the space of any irreducible framed link in the 3-sphere, modulo rotations. This leads us to the homotopy type of the space of any knot in the solid torus, thus answering a question posed by Arnold. We si