Journal of Knot Theory and Its Ramifications

Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

We give a formula for the Casson knot invariant of a p-periodic knot in S³ whose quotient link is a 2-bridge link with Conway's normal form C(2, 2n₁, -2, 2n₂, …, 2n_2m, 2) via the integers p, n₁, n₂, …, n_2m(p ≥ 2 and m ≥ 1). As an application, for any integers n₁, n₂, ≥, n_2m with the same sign, we determine the Δ-unknotting number of a p-periodic knot in S³ whose quotient is a 2-bridge link C(2, 2n₁, -2, 2n₂, ≥, 2n_2m, 2) in terms of p, n₁, n₂, ≥, n_2m. In addition, a recurrence formula for calculating the Alexander polynomial of the 2-bridge knot with Conway's normal form C(2n₁, 2n₂, ≥, 2n_m) via the integers n₁,n₂, ≥, n_m is included.

In this paper, we show how to realize all knot (and link) types as smooth curves of constant curvature. Our proof is constructive: we build the knots with copies of a fixed finite number of "building blocks" that are particular segments of helices and circles. We use these building blocks to construct all closed braids.

A flow in three-dimensions is universal if the periodic orbits contains all knots and links. Universal flows were shown to exist by Ghrist, and can be constructed by means of templates. Likewise, a planar diffeomorphism is universal if it has a suspension flow which is a universal flow. In this paper we prove the existence of a homoclinic trellis type for which any representative diffeomorphism is universal. This trellis type is remarkable in that it has zero entropy, and only two homoclinic intersection points.

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

In the article we study the multi-variable Alexander polynomial of a link in a plat form or in a closed braid form. By using the method, we find an algorithm how to compute the multi-variable Alexander polynomial of the 2a-fold dihedral cover and the a-fold irregular cover of a two bridge knot K(a,b).