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Virtual knot groups and almost classical knots

Tom 238 / 2017

Hans U. Boden, Robin Gaudreau, Eric Harper, Andrew J. Nicas, Lindsay White Fundamenta Mathematicae 238 (2017), 101-142 MSC: Primary 57M25; Secondary 57M27. DOI: 10.4064/fm80-9-2016 Opublikowany online: 6 February 2017

Streszczenie

We define a group-valued invariant $\overline G_K$ of virtual knots $K$ and show that $VG_K=\overline G_K \mathbin{*_{\mathbb Z}\,} \mathbb Z^2,$ where $VG_K$ denotes the virtual knot group introduced by Boden et al. We further show that $\overline G_K$ is isomorphic to both the extended group $EG_K$ of Silver–Williams and the quandle group $QG_K$ of Manturov and Bardakov–Bellingeri.

A virtual knot is called almost classical if it admits a diagram with an Alexander numbering, and in that case we show that $\overline G_K$ splits as $G_K*\mathbb Z$, where $G_K$ is the knot group. We establish a similar formula for mod $p$ almost classical knots and derive obstructions to $K$ being mod $p$ almost classical.

Viewed as knots in thickened surfaces, almost classical knots correspond to those that are homologically trivial. We show they admit Seifert surfaces and relate their Alexander invariants to the homology of the associated infinite cyclic cover. We prove the first Alexander ideal is principal, recovering a result first proved by Nakamura et al. using different methods. The resulting Alexander polynomial is shown to satisfy a skein relation, and its degree gives a lower bound for the Seifert genus. We tabulate almost classical knots up to six crossings and determine their Alexander polynomials and virtual genus.

Autorzy

  • Hans U. BodenMathematics & Statistics
    McMaster University
    Hamilton, Ontario L8S 4K1, Canada
    e-mail
  • Robin GaudreauMathematics & Statistics
    McMaster University
    Hamilton, Ontario L8S 4K1, Canada
    e-mail
  • Eric HarperMathematics & Statistics
    McMaster University
    Hamilton, Ontario L8S 4K1, Canada
    e-mail
  • Andrew J. NicasMathematics & Statistics
    McMaster University
    Hamilton, Ontario L8S 4K1, Canada
    e-mail
  • Lindsay WhiteMathematics & Statistics
    McMaster University
    Hamilton, Ontario L8S 4K1, Canada
    e-mail

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