Journal of Knot Theory and Its Ramifications

It is shown that two knots can be transformed into each other by C_n-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a C_n-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a C_n-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

If is a branched covering between closed surfaces, there are several easy relations one can establish between the Euler characteristics and χ(Σ), orientability of Σ and , the total degree, and the local degrees at the branching points, including the classical Riemann–Hurwitz formula. These necessary relations have been shown to be also sufficient for the existence of the covering except when Σ is the sphere 𝕊 (and when Σ is the projective plane, but this case reduces to the case Σ = 𝕊). For Σ = 𝕊 many exceptions are known to occur and the problem is widely open.

Generalizing methods of Baránski, we prove in this paper that the necessary relations are actually sufficient in a specific but rather interesting situation. Namely under the assumption that Σ = 𝕊, that there are three branching points, that one of these branching points has only two pre-images with one being a double point, and either that and that the degree is odd, or that has genus at least one, with a single specific exception. For the case of we also show that for each even degree there are precisely two exceptions.

Imposing different conditions on minimality of reduced braid words and new criteria on their representatives, we define braid family representatives and establish one-to-one correspondence with families of knots and links given in Conway notation.

Apart from their generic relationship to knots and their application to particle physics [1], flattened Moebius strips (FMS) are of intrinsic interest as elements of a genus with specific rules of combination and a unique taxonomy. Here, FMS taxonomy is developed in detail from combinatorial and lexicographic points of view which include notions of degeneracy, completeness and excited states. The results are compared to the standard, spin-parameterized, abstract hierarchy derived by group-theoretic arguments as the direct product of vector spin spaces [2]. A review of the notion of excited states then leads to a new and different model of Beta decay that employs only fusion and fission. There is additional discussion of the relationship between twist and charge and an operator/tensor formulation of the fusion and fission of basic FMS units. Associating a Hopf algebra to FMS operations as a step toward a topological quantum field theory is also investigated. The notion of spinor/twistor networks is seen to emerge from a consideration of FMS configurations for higher values of twist and the introduction of a mode dual to the canonical FMS configuration. The last section discusses the connection of the MS genus to fiber bundle/gauge theory, the concept of spin, and the Dirac equation of the electron.

We express the first non-vanishing Massey–Milnor linkings in terms of Chern–Simons–Witten configuration space integrals in perturbative quantum field theory.