Maps from knots in the cylinder to flat-virtual knots

V. O. Manturov; I. M. Nikonov

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arXiv · 2210.09689

Maps from knots in the cylinder to flat-virtual knots

V. O. Manturov·I. M. Nikonov

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Abstract

In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat knots. The main construction takes a diagram on a cylinder (torus) and adds some ``invisible'' crossings which gives rise to a diagram which can be formally immersed but not embedded (drawn) on the cylinder (torus) and living comfortably in thickened surfaces of higher genera. This allows one to ``pull back'' invariants of virtual theory to the theory of knots in the thickened cylinder (torus) where the parity bracket and other picture-valued invariants are not strong enough.

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BibTeX RIS

V. O. Manturov, I. M. Nikonov. 2022-10-18. Maps from knots in the cylinder to flat-virtual knots. https://arxiv.org/abs/2210.09689

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