Link invariants derived from multiplexing of crossings

Abstract

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers $m_i$ $(i=1,\dots ,n)$ and an ordered $n$-component virtual link diagram $D$, a new virtual link diagram $D(m_1,\dots ,m_n)$ is obtained from $D$ by the multiplexing of all crossings. For welded isotopic virtual link diagrams $D$ and $D^{\prime }$, $D(m_1,\dots ,m_n)$ and $D^{\prime }(m_1,\dots ,m_n)$ are welded isotopic. From the point of view of classical link theory, it seems very interesting that $D(m_1,\dots ,m_n)$ could not be welded isotopic to a classical link diagram even if $D$ is a classical one, and new classical link invariants are expected from known welded link invariants via the multiplexing of crossings.

9 pages, 16 figures

References in corpus (3)

• Arrow calculus for welded and classical links
• The 2-generalized knot group determines the knot
• Generalised knot groups distinguish the square and granny knots (with an appendix by David Savitt)

Cited by in corpus (1)

• Generalized virtualization on welded links