We investigate when two virtual knots can be related by Δ-moves. We show that the L-polynomials are finite type invariants of order one. As a consequence, they are invariant under Δ-moves for any oriented virtual knot. As an application, we study virtual knots that can be converted into the trivial knot or the virtual knot 2.1 by a sequence of Δ-moves.
We study Fox colorings of tangle diagrams by [Formula: see text] or [Formula: see text], where [Formula: see text] is odd. For an [Formula: see text]-colored [Formula: see text]-string tangle diagram, the colors at the [Formula: see text] boundary points form a vector [Formula: see text]. We show that a vector [Formula: see text] arises from an [Formula: see text]-colored classical tangle diagram if and only if the alternating sum [Formula: see text] vanishes. We then determine how this restriction changes for virtual tangle diagrams. For [Formula: see text], the realizability of a vector [Formula: see text] is characterized by a divisibility condition on [Formula: see text]. For [Formula: see text], every vector [Formula: see text] is realizable by a virtual tangle diagram.
Knotted 4-regular graphs appear in quantum gravity models as possible states of geometry, since these arise as graphs dual to certain triangulations of three-dimensional spaces. In this paper, methods to enumerate these graphs are discussed. We develop a new graph invariant, which is used to obtain a census of knotted 4-regular graphs represented by Dowker-Thistlethwaite sequences of five nodes or less, with graphs and their mirror images identified in the census. This invariant is used to place a lower limit on the minimal number of nodes for any diagram of a graph.
In this paper, we correct a chirality error in Proposition 4.5 and the resulting amicable pair in Proposition 5.4. The corrected pair consists of the right-handed trefoil [Formula: see text] and the left-handed torus knot [Formula: see text], rather than [Formula: see text] and [Formula: see text]. We give corrected statements and proofs, clarify the proof of the torus-knot inclusion in Proposition 5.1, and record a direct consequence: [Formula: see text] and [Formula: see text] are proper amicable knots for all [Formula: see text].
[Formula: see text]A braiding of a branched cover of the 3-sphere that is branched over a knot is a continuous map of the cover into the product of the sphere with a two-dimensional disk that has the property that the projection onto the sphere factor induces the covering. A folding maps the cover into the product of a sphere with an interval. Moreover, away from the branch set, the map is a general position immersion. Cyclic branched covers can be braided so that the map is an embedding when the disk factor is two-dimensional. Dihedral branched covers can be folded. [Formula: see text]As far as possible, the braidings and foldings that are presented are quite detailed. In particular, the paper focuses upon a folding of the dihedral cover of the 3-sphere that is branched along a torus knot of type (2,[Formula: see text]5). The cover also is homeomorphic to the 3-sphere.
