The L2L^2-Alexander invariant detects the unknot

In this article, we present some of the properties of the $L^2$-Alexander invariant of a knot defined by Li and Zhang, some of which are similar to those of the classical Alexander polynomial. Notably we prove that the $L^2$-Alexander invariant detects the trivial knot.Comment: Some typos were corrected, and we added formulas for the invariant of the inverse knot or the mirror image of a kno

Geometric triangulations and the Teichm\"uller TQFT volume conjecture for twist knots

We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichm\"uller TQFT and to prove the associated volume conjecture for all twist knots, using the saddle point method.Comment: v4: 90 pages, 25 figures. Comments welcome. Some of the results in this paper were announced in a note at the C. R. Acad. Sci. Paris, and some were detailed in the arXiv v1. Since v3, we made minor corrections, we added details in Section 2.9, and we added Remark 3.

Markov moves, L^2-Burau maps and Lehmer's constants

We study the effect of Markov moves on L^2-Burau maps of braids, in order to construct link invariants from these maps with a process mirroring the well-known Alexander-Burau formula. We prove such a Markov invariance for the L^2-Burau maps which descend to the groups of the braid closures or lower, and for which the associated link invariants are twisted L^2-Alexander torsions. When the L^2-Burau map descends to a link group, the corresponding link invariant was known to be the L^2-Alexander torsion of the link by a previous result of A. Conway and the author. Furthermore, we find two counter-examples to Markov invariance, meaning two families of L^2-Burau maps that cannot yield link invariants with the process described in our paper. The proofs use relations between Fuglede-Kadison determinants, Mahler measures, and random walks on Cayley graphs, as well as works of Boyd, Bartholdi and Dasbach-Lalin. Along the way, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups. As a consequence, we partially answer a question of Lück, as we provide new upper bounds for Lehmer's constants for all torsionfree groups which have non-cyclic free subgroups. Our results suggest that twisted L^2-Alexander torsions are the only link invariants we can hope to construct from L^2-Burau maps with the present approach

Workshop "Fictionalization in Math outreach"

From the historical figures met in Doctor Who to the scientific hijinks of the kids in the Magic School Bus, most sciences have had their time in the spotlight in fictionalized works. What about math? How can we do mathematics outreach through fiction? But first… should we? Is a western the best setting to present knot theory? What can romance stories tell us about the axiom of choice? Along the way of discussing such questions, we may also review mathematical models of non-linear stories like gamebooks, and we will try to create new math activities. This workshop is a story you cannot miss, a story in which YOU are the (co)protagonist!         Some first questions: For a given mathematical subject to popularize is it relevant to present it through fiction? If we don’t know if it is relevant, how could we determine this? If it seems relevant, which type of fiction could be appropriate?  Proposed plan of approach :  Reviewing forms of fiction: novel, comic book, radio, theatre, movie, board game… Reviewing genres of fiction: science fiction, comedy, historical, mystery, horror…  Linearity of the plot and influence of the audience: anything between an un-modifiable classical story and a full improve show with audience input.  Non-linear plots and graph theory, with the example of « choose your own adventure » gamebooks.  Create a new outreach activity mixing mathematics and creative writing: present basics of graph theory and non-linear stories, then make the audience create a non-linear story associated to a given graph