Alexander-Markov correspondence for doodles on closed surfaces

TL;DR

The paper develops a twisted virtual doodle framework on closed surfaces and its braid-theoretic counterpart via twisted virtual twin groups $TVT_n$ and the pure subgroups $PTVT_n$. It proves an Alexander-type theorem equating doodle closures with twisted virtual twins and a Markov-type theorem characterizing closure equivalence, then provides explicit presentations and structural decompositions for $PTVT_n$ and $TVT_n$, including RAAG structure, semidirect decompositions, and finiteness properties. These results unify classical and virtual doodle theories, offering algebraic tools for classification and paving the way for representations and further structural analysis. The findings demonstrate trivial centers and residual finiteness/Hopfian properties, with detailed decompositions revealing the internal architecture of twisted virtual twin groups. Overall, the work extends braid-theoretic methods to twisted, non-orientable settings and deepens the connection between surface doodles and planar diagrammatic techniques.

Abstract

In this paper, we introduce twisted virtual doodles, defined as stable equivalence classes of immersed circles on closed surfaces that may be non-orientable. These objects admit planar representative diagrams, considered up to a suitable set of Reidemeister-type moves. To develop the associated braid-theoretic framework, we define twisted virtual twin groups as natural extensions of virtual twin groups, and establish Alexander- and Markov-type theorems in this set-up. This shows that twisted virtual doodles unify and extend both classical and virtual doodle theories. We further investigate the structure of the pure twisted virtual twin group, providing a presentation and deriving several structural and combinatorial properties. In particular, we obtain two interesting decompositions of the twisted virtual twin group and its pure subgroup, from which it follows that both groups have trivial center and are residually finite as well as Hopfian.

Figures (14)

• Figure 3: Examples of trivial twisted virtual doodles on surfaces.
• Figure 4: Example of a non-trivial twisted virtual doodle on a surface.
• Figure 5: Examples of twisted virtual doodle diagrams.
• Figure 6: Extended Reidemeister moves for twisted virtual doodle diagrams.
• Figure 7: Attaching a band and a twisted band.

Theorems & Definitions (56)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Proposition 2.8
• proof
• Definition 2.9