In this paper, we introduce a new tool called lassos, which we use to establish a correspondence between cellular link diagrams on closed surfaces and equivalence classes of virtual link diagrams — this is a diagrammatic analog of a well-known correspondence between the links represented by these diagrams. Under these correspondences, the traditional notion of primeness for virtual links is stricter than the one for links in thickened surfaces. Our main result, generalizing a classical result of Menasco, is that an alternating link in a thickened surface is prime in the stricter sense unless it is “obviously” composite. (Adams et al. and Howie–Purcell previously extended Menasco’s result for the other notion of primeness.)
In the present paper, a polynomial invariant [Formula: see text] for virtual knots is constructed by 0-smoothing at all real crossings and using the polynomial invariant [Formula: see text] for flat virtual knots. We further discuss some properties of the polynomial invariant [Formula: see text]. Finally, we construct a family of diagrams for virtual knots which can be distinguished from each other by invariant [Formula: see text].
We consider the representation [Formula: see text] which extends the Lawrence-Bigelow-Krammer representation from the braid group [Formula: see text] to the virtual braid group [Formula: see text] on 3 strands. After specializing parameters [Formula: see text], [Formula: see text] to non-zero complex numbers, we study the irreducibility of [Formula: see text]. We prove that [Formula: see text] is irreducible for all non-zero complex numbers [Formula: see text], [Formula: see text] except for the case [Formula: see text].
In this paper, we give Bennequin–Plamenevskaya–Shumakovitch-type lower bounds for the concordance invariant [Formula: see text] introduced by Kronheimer and Mrowka. The proof is a consequence of computations for torus knots and the cobordism inequality of [Formula: see text] due to Gong, combined with well-known arguments used for slice-torus invariants.
For a knot [Formula: see text], let [Formula: see text] denote the set of nontrivial knot types represented by simple closed curves on a minimal genus Seifert surface of [Formula: see text]. We study the relation [Formula: see text] and its symmetric part, which leads to the notion of amicable knots: Knots [Formula: see text] and [Formula: see text] are called amicable if each is represented by a simple closed curve on a minimal genus Seifert surface of the other. A classical result of Lyon implies that the family of torus knots is universal for this realization problem: For every nontrivial knot type [Formula: see text], there exists a torus knot [Formula: see text] such that [Formula: see text]. In contrast, one of the main results of this paper is that no single knot is universal: For every knot [Formula: see text], there exists a knot [Formula: see text] such that [Formula: see text]. We also study explicit examples, keeping track of chirality throughout. Writing [Formula: see text] and [Formula: see text] for the right-handed positive torus knots, we show that [Formula: see text] and [Formula: see text] are amicable, whereas [Formula: see text] and the figure-eight knot [Formula: see text] are not. We also describe the hosting sets of both chiralities of the trefoil in terms of primitive slope classes on their once-punctured torus fibers.
