# Concordances to prime hyperbolic virtual knots

## TL;DR

This paper proves that every virtual knot in a thickened surface is concordant to a prime satellite and hyperbolic knot, extending classical results and addressing new challenges in the virtual setting.

## Contribution

It introduces the concept of complementary tangles and demonstrates that virtual knots can be simplified to prime hyperbolic or satellite knots while preserving the Alexander polynomial in certain cases.

## Key findings

- Every virtual knot is concordant to a prime satellite knot.
- Every virtual knot is concordant to a prime hyperbolic knot.
- The Alexander polynomial is preserved for homologically trivial knots during this process.

## 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.

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Source: https://tomesphere.com/paper/1904.05288