Shinji Fukuhara

ENERGY OF A KNOT

Publisher Summary This chapter discusses energy of a knot. It presents a method to deform a knot canonically and obtain its standard form. A normalized polygonal knot means a polygonal knot such that length of every edge is 1. If a knot is set in viscous liquid, which absorbs kinetic energy, then the knot will move and gradually reduce its electric energy. After some time, its electric energy will become minimal and a standard form of the previous knot will be obtained. Thus, a nonelastic string, with electrons distributed on it, can be made.

Generalized Dedekind symbols associated with the Eisenstein series

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.

Period polynomials and explicit formulas for Hecke operators on Γ 0 (2)

Let Sw+2( 0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup 0(N). We first determine explicit formulas for period polynomials of elements in Sw+2( 0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2( 0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to 0(2). This implies that there are natural correspondences between the spaces of cusp forms on 0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2( 0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2( 0(2)).

Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

Abstract Let be the vector space of cusp forms of weight w + 2 on the full modular group, and let denote its dual space. Periods of cusp forms can be regarded as elements of . The Eichler-Shimura isomorphism theorem asserts that odd (or even) periods span . However, periods are not linearly independent; in fact, they satisfy the Eichler-Shimura relations. This leads to a natural question: which periods would form a basis of . First we give an answer to this question. Passing to the dual space , we will determine a new basis for . The even period polynomials of this basis elements are expressed explicitly by means of Bernoulli polynomials. Next we consider three spaces— , the space of even Dedekind symbols of weight w with polynomial reciprocity laws, and the space of even period polynomials of degree w. There are natural correspondences among these three spaces. All these spaces are equipped with compatible action of Hecke operators. We will find explicit forms of period polynomials and the actions of Hecke operators on the period polynomials. Finally we will obtain explicit formulas for Hecke operators on in terms of Bernoulli numbers Bk and divisor functions σk (n), which are quite different from the Eichler-Selberg trace formula.

Explicit formulae for two-bridge knot polynomials

A two-bridge knot (or link) can be characterized by the so-called Schubert normal form K p;q where p and q are positive coprime integers. Associated to K p;q there are the Conway polynomial rK p;q .z/ and the normalized Alexander polynomial1K p;q.t/. However, it has been open problem how rK p;q.z/ and 1K p;q.t/ are expressed in terms of p and q. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions in p and q.

Hecke operators on weighted Dedekind symbols

Abstract Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke eigenforms. In particular we give explicit formulae for generalized Ramanujans tau functions.

Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter r having positive imaginary part. When T → i∞, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable r. We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).

Dedekind symbols associated with J-forms and their reciprocity law

Abstract We define a Dedekind symbol associated with a J-form (J-forms generalize the usual Jacobi forms). Then we prove the reciprocity law for Dedekind symbols. As an example, we give an explicit description for the Dedekind symbol associated with Weierstrass ℘ -function. The reciprocity law in this case then yields new trigonometric identities. This in turn gives rise to a “generating function” of Apostol reciprocity law for the generalized Dedekind sums.

Kauffman–Jones polynomial of a curve on a surface

We introduce a Kauffman–Jones type polynomial ℒγ(A) for a curve γ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial ℒγ(A) is a Laurent polynomial in one variable A and is an invariant of the homotopy class of γ. As an application, we obtain an estimate in terms of the span of ℒγ(A) for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of ℒγ(A) and show some computational results on the span of ℒγ(A).

NON-COMMUTATIVE POLYNOMIAL RECIPROCITY FORMULAE

We prove non-commutative reciprocity formulae for certain polynomials using Foxs free differential calculus. The abelianizations of these reciprocity formulae rediscover the polynomial reciprocity formulae of Carlitz and Berndt–Dieter. Further, many other reciprocity formulae related to Dedekind sums are rederived from our polynomial reciprocity formulae; these include, for instance, generalizations of the classical Eisenstein reciprocity formula.