Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial
By Dror Bar-Natan, Zsuzsanna Dancso. 2014
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This is the first in a series of papers studying w-knots, and more generally, w-knotted objects (w-braids, w-tangles, etc). These are classes of knotted objects which are wider, but weaker than their “usual” counterparts. The group of w-braids was studied (under the name “welded braids”) by Fenn, Rimanyi and Rourke and was shown to be isomorphic to the McCool group of “basisconjugating” automorphisms of a free group Fn : the smallest subgroup of Aut.Fn/ that contains both braids and permutations. Brendle and Hatcher, in work that traces back to Goldsmith, have shown this group to be a group of movies of flying rings in R 3 . Satoh studied several classes of w-knotted objects (under the name “weaklyvirtual”) and has shown them to be closely related to certain classes of knotted surfaces in R 4 . So w-knotted objects are algebraically and topologically interesting. Here we study finite-type invariants of w-braids and w-knots. Following Berceanu and Papadima, we construct homomorphic universal finite-type invariants of w-braids. The universal finite-type invariant of w-knots is essentially the Alexander polynomial. Much as the spaces A of chord diagrams for ordinary knotted objects are related to metrized Lie algebras, the spaces A w of “arrow diagrams” for w-knotted objects are related to not-necessarily-metrized Lie algebras. Many questions concerning w-knotted objects turn out to be equivalent to questions about Lie algebras. Later in this paper series we re-interpret the work of Alekseev and Torossian on Drinfel’d associators and the Kashiwara‐Vergne problem as a study of w-knotted trivalent graphs. 57M25, 57Q45
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/c6d658e865b937c8c5412b6a3820e4d0281b5f05
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- A journey through loop braid groups (2016, methodology): “Other equivalent definitions of virtual braid groups have been introduced for instance in [7, 34,49].”