Journal of Knot Theory and Its Ramifications

Two recent publications describe realizable Gauss diagrams using conditions stating that the number of chords in certain sets of chords is even or odd. We demonstrate that these descriptions are incorrect by finding multiple counter-examples. However, the idea of having a parity-based description of realizable Gauss diagrams is attractive. We recall that realizability of Gauss diagrams as touch curves can be described via bipartite graphs. We show that realizable Gauss diagrams can be described via bipartite graphs.

In this paper, we find that for each given spatial graph, the number of its maximal trees is finite, and the quotient space corresponding to some maximal tree is also a spatial graph and it is homeomorphic to a disjoint union of wedge sum of circles. That means this quotient space is orientable and colorable. Thus, we define two equivalence invariants of spatial graphs, the coloring number bracket and the Alexander invariant set, which are coming from the coloring invariant and Alexander matrix.

We consider cyclic branched coverings of a 3-parameter family of rational knots in $S^3$ and study the left orderability of their fundamental groups. We first compute the nonabelian ${\mathrm{\text{SL}}}_2(ℂ)$-character varieties of the rational knots $C(2p,2m,2n+1)$ in the Conway notation, where $p,m,n$ are integers. We then study real points on these varieties and finally use them to determine the left orderability of the fundamental groups of cyclic branched coverings of $C(2p,2m,2n+1)$.

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic’s continued fraction expansion. Consequently, we give a volume’s upper bound for some sequences of Lorenz knots complements, linearly in terms of the corresponding braid index.

Also, for any punctured hyperbolic surface we give volume’s bounds for the canonical lift complement relative to some sequences of sets of closed geodesics in terms of the geodesics length.

We extend our previous work on biquandle module invariants of oriented surface-links to the case of unoriented surface-links using bikei modules. The resulting infinite family of enhanced invariants proves be effective at distinguishing unoriented and especially non-orientable surface-links; in particular, we show that these invariants are more effective than the bikei homset cardinality invariant alone at distinguishing non-orientable surface-links. Moreover, as another application we note that our previous biquandle modules which do not satisfy the bikei module axioms are capable of distinguishing different choices of orientation for orientable surface-links as well as classical and virtual links.

In this paper, we present necessary and sufficient combinatorial conditions for a link to be projective, that is, a link in $ℝ{\mathbb{P}}^3$. This characterization is closely related to the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the notion of symmetric cycle, an interesting issue arising in projective links leading us to an easy necessary condition for alternating anti-inversely symmetric links.

Although the connected sum of trivial knots is trivial in classical knot theory, there is a nontrivial connected sum of trivial knots in virtual knot theory. In this paper, we give connected sum formulae of the intersection polynomials of virtual knots introduced in the preceding paper [6]. As an application, we show that there are infinitely many connected sums of any pair of virtual knots.

This paper discusses some geometric ideas associated with knots in real projective 3-space $ℝP^3$. These ideas are borrowed from classical knot theory. Since knots in $ℝP^3$ are classified into three disjoint classes: affine, class-$0$ non-affine and class-$1$ knots, it is natural to wonder in which class a given knot belongs to. In this paper we attempt to answer this question. We provide a structure theorem for these knots which helps in describing their behavior near the projective plane at infinity. We propose a procedure called space bending surgery, on affine knots to produce several examples of knots. We later show that this operation can be extended on an arbitrary knot in $ℝP^3$. We then study the notion of companionship of knots in $ℝP^3$ and using it we provide geometric criteria for a knot to be affine. We also define a notion of “genus” for knots in $ℝP^3$ and study some of its properties. We prove that this genus detects knottedness in $ℝP^3$ and gives some criteria for a knot to be affine and of class-$1$. We also prove a “non-cancellation” theorem for space bending surgery using the properties of genus. Then we show that a knot can have genus 1 if and only if it is a cable knot with a class-1 companion. We produce examples of class-$0$ non-affine knots with genus $1$. Thus we highlight that, $ℝP^3$ admits a knot theory with a truly different flavor than that of $S^3$ or ${ℝ}^3$.