Journal of Knot Theory and Its Ramifications

In this paper, following the scheme of [Borromean rings and linkings, J. Geom. Phys.60 (2010) 823–831; Combinatoric and diagrammatic study in knot theory, J. Knot Theory Ramifications16 (2007) 1235–1253; Massey–Milnor linking = Chern–Simons–Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903], we study the first non-vanishing self-linkings of knots, aiming at the study of combinatorial formulae and diagrammatic representation. The upshot of perturbative quantum field theory is to compute the Feynman diagrams explicitly, though it is impossible in general. Along this line in this paper we could not only compute some Feynman diagrams, but also give the explicit and combinatorial formulae.

We introduce invariants of flat virtual links which are induced from Vassiliev invariants of degree one for virtual links. Also we give several properties of these invariants for flat virtual links and examples. In particular, if the value of some invariants of flat virtual knots F are non-zero, then F is non-invertible so that every virtual knot overlying F is non-invertible.

We consider a graph link L in S³ which admits a symmetry of the exterior given by a boundary component preserving periodic homeomorphism f (orientation-preserving or not). This paper shows that L is obtained by a specific construction method if f is not a strong inversion. As an application, we provide a complete classification of 2-component graph links with symmetries of the exteriors which are not extendable to symmetries of S³.

It is known that every nontrivial knot has at least two quadrisecants. Given a knot, we mark each intersection point of each of its quadrisecants. Replacing each subarc between two nearby marked points with a straight line segment joining them, we obtain a polygonal closed curve which we will call the quadrisecant approximation of the given knot. We show that for any hexagonal trefoil knot, there are only three quadrisecants, and the resulting quadrisecant approximation has the same knot type.

For a knot K with a framing F, we show that the knot rack R(K, F) of (K, F) has the structure of a rack extension of the knot quandle Q(K), via the projection canonically obtained by disregarding its framing. From this fact, we can construct a cohomology class [ϕ_F] of Q(K) corresponding to the framing: a cohomology class in for K non-trivial, or one in for K trivial, where w_F denotes the writhe of (K, F). Moreover, when K is non-trivial, [ϕ_F] has a natural decomposition to the sum of and , where [ψ] is independent to F. This correspondence gives a bijection between the set of framings of K and .

It is known that a maximal collection of essential annuli in H₂ could contain exactly 1, or 2, or at most 3 annuli, and it is clear how they are embedded in H₂. It is also known that for a maximal collection of essential annuli in H_n with n ≥ 3, , and the bounds are sharp. Moreover, for each integer m, 2 ≤ m ≤ 4n - 5, there exists a maximal collection of essential annuli in H_n which contains exactly m annuli. In this paper, we detect how a maximal collection of essential annuli in H_n with n ≥ 3 are embedded in H_n.

The subject is a localized disturbance in the form of a torus knot of an otherwise featureless continuum. The knot's topologically quantized, self-sustaining nature emerges in an elementary, straightforward way on the basis of a simple geometric model, one that constrains the differential geometric basis it otherwise shares with General Relativity (GR). Two approaches are employed to generate the knot's solitonic nature, one emphasizing basic differential geometry and the other based on a Lagrangian. The relationship to GR is also examined, especially in terms of the formulation of an energy density for the Lagrangian. The emergent knot formalism is used to derive estimates of some measurable quantities for a certain elementary particle model documented in previous publications. Also emerging is the compatibility of the torus knot formalism and, by extension, that of the cited particle model, with general relativity as well as with the Dirac theoretic notion of antiparticles.

We extend the notion of geometric intersection numbers 0 and 1, for circles in the Dehn quandle, to general quandles, assuming existence of elements having certain algebraic properties in the latter. These properties enable the construction of many useful 2 and 3 cycles in Dehn quandle homology. Particularly, we construct a special 2-cycle homology representative and an operation which promotes it to higher dimensional cycles of the same type, in the Dehn quandle, and show how to do so in any quandles admitting such elements. For such quandles, this gives some non-triviality results in the quandle homology. We look at a secondary rack formed by tuples of elements of the Dehn quandle and using the ideas above, obtain some basic non-triviality results in its homology as well. We also give some initial possibilities for its application in representing monodromy of singularities of elliptic fibrations.