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Register a new userOn polynomial Torus Knots
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Authors
Pierre-Vincent Koseleff
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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.
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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 $K_{2,N}$.
In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem.
We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial.
We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C].
We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
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