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The beta function of a knot. (English) Zbl 0972.57006

The author studies a metric invariant of a smooth knot \(K\) embedded in the standard 3-sphere. It is defined as a double integral using the parametrization \(x\to\gamma(x)\) of \(K\) by the arc-length parameter \(x\): \[ B_K (s)= \iint_{[0,\ell] \times[0,\ell]} \bigl\|\gamma (x)-\gamma(y) \bigr \|^sdx dy \] where \(dx\) and \(dy\) denote the arc-lengths on the two copies of \(K\), and \(\ell\) is the total length of \(K\). Of course, the function \((x,y)\to \|\gamma (x)-\gamma(y) \|^s\) is continuous for \(\text{Re}(s)>0\), so that the integral converges in that domain. The main result of the paper states that the function \(s\to B_K(s)\) extends analytically to a meromorphic function of a complex variable, with only possible poles at \(-1,-3,-5,\dots\). The author determines the first residues of that meromorphic function. The residue at \(-2j-1\) is of the form \(\int_KP_j(k, \tau)dx\), where \(P_j\) is an explicitly computable polynomial in the curvature \(k\), the torsion \(\tau\) and their derivatives.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
53C40 Global submanifolds
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
33B15 Gamma, beta and polygamma functions
53A04 Curves in Euclidean and related spaces
33E20 Other functions defined by series and integrals
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References:

[1] Bernstein J., Funct. Anal. Appl. 6 pp 272– (1972)
[2] DOI: 10.2307/2946626 · Zbl 0817.57011
[3] DOI: 10.1080/10586458.1993.10504264 · Zbl 0818.57007
[4] DOI: 10.1016/0040-9383(91)90010-2 · Zbl 0733.57005
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