• DOI:10.1142/S021821651350079X
• Corpus ID: 119308260

Cubulated moves and discrete knots

@article{Hinojosa2013CubulatedMA,
  title={Cubulated moves and discrete knots},
  author={Gabriela Hinojosa and Alberto Verjosvky and Cynthia Verjovsky Marcotte},
  journal={arXiv: Geometric Topology},
  year={2013}
}

• Gabriela Hinojosa, Alberto Verjosvky, C. Marcotte
• Published 8 February 2013
• Mathematics
• arXiv: Geometric Topology

In this paper, we prove than given two cubic knots $K_1$, $K_2$ in $\mathbb{R}^3$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister moves for classical tame knots. We use the fact that a cubic knot is determined by a cyclic permutation of contiguous vertices of the $\mathbb{Z}^3$-lattice (with some restrictions), to describe some of the classic invariants and properties of the knots in terms of such cyclic permutations, by projecting onto a plane such that it is injective when restricted to the $\mathbb{Z}^3$-lattice and the image of the $\mathbb{Z}^3$-lattice is dense.