Concordances to prime hyperbolic virtual knots

Abstract

Let $$\Sigma _0,\Sigma _1$$ be closed oriented surfaces. Two oriented knots $$K_0 \subset \Sigma _0 \times [0,1]$$ and $$K_1 \subset \Sigma _1 \times [0,1]$$ are said to be (virtually) concordant if there is a compact oriented 3-manifold W and a smoothly and properly embedded annulus A in $$W \times [0,1]$$ such that $$\partial W=\Sigma _1 \sqcup -\Sigma _0$$ and $$\partial A=K_1 \sqcup -K_0$$ . This notion of concordance, due to Turaev, is equivalent to concordance of virtual knots, due to Kauffman. A prime virtual knot, in the sense of Matveev, is one for which no thickened surface representative $$K \subset \Sigma \times [0,1]$$ admits a nontrivial decomposition along a separating vertical annulus that intersects K in two points. Here we prove that every knot $$K \subset \Sigma \times [0,1]$$ is concordant to a prime satellite knot and a prime hyperbolic knot. For homologically trivial knots in $$\Sigma \times [0,1]$$ , we prove this can be done so that the Alexander polynomial is preserved. This generalizes the corresponding results for classical knot concordance, due to Bleiler, Kirby–Lickorish, Livingston, Myers, Nakanishi, and Soma. The new challenge for virtual knots lies in proving primeness. Contrary to the classical case, not every hyperbolic knot in $$\Sigma \times [0,1]$$ is prime and not every composite knot is a satellite. Our results are obtained using a generalization of tangles in 3-balls we call complementary tangles. Properties of complementary tangles are studied in detail.