Journal of Knot Theory and Its Ramifications

We construct a multi-variable polynomial invariant for virtual knots and links via the concept of a decorated virtual magnetic graph diagram. The invariant is a generalization of the Jones–Kauffman polynomial for virtual knots and links. We show some features of the invariant including an evaluation of the virtual crossing number of a virtual knot or link.

In the present paper, we extend the "torus gauge fixing" approach by Blau and Thompson, which was developed in [10] for the study of Chern–Simons models with base manifolds M of the form M = Σ × S¹, in a suitable way. We arrive at a heuristic path integral formula for the Wilson loop observables associated to general links in M. We then show that the right-hand side of this formula can be evaluated explicitly in a non-perturbative way and that this evaluation naturally leads to the face models in terms of which Turaev's shadow invariant is defined.

In this essay, we present our interpretation of the work of P. Tait and M. Haseman on achiral alternating knots. We have tried to elucidate the meaning of several concepts used by them such as: Amphicheirals, skew-amphicheirals, 1st or 2nd order, 1st or 2nd class, distorsions, etc. We also comment on some of their many pictures.

The image of a polygonal knot K under a spherical inversion of ℝ³ ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon . This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group.

The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ³. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ³. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.

Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ³ and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ³ and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge.

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and of Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with Spin^c-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-dimensional handlebody is homotopic to a complex one.