Welded Links

Meridional rank and bridge numbers of knotted surfaces and welded knots

October 19, 2021October 19, 2021 | chrisman.76

Jason Joseph (Rice University)

The meridional rank conjecture (MRC) posits that the meridional rank of a classical knot is equal to its bridge number. In this talk we investigate whether or not this is a reasonable conjecture for knotted surfaces and welded knots. In particular, we find criteria to establish the equality of these values for several large families. On the flip side, we examine the behavior of meridional rank of knotted spheres under connected sum, and, using examples first studied by Kanenobu, show that any value between the theoretical limits can be achieved. This means that either the MRC is false for knotted spheres, or that their bridge number fails to be (-1)-additive. This is joint work with Puttipong Pongtanapaisan.

Welded Tangles and the Kashiwara-Vergne Groups

September 21, 2021September 20, 2021 | chrisman.76

Zsuzsanna Dancso (The University of Sydney)

In this talk I will explain a general method of “translating” between a certain type of problem in topology, and solving equations in graded spaces in (quantum) algebra. I’ll talk through several applications of this method from the 90’s to today: Drinfel’d associators and parenthesised braids, Grothendieck-Teichmuller groups, welded tangles and the Alekseev-Enriquez-Torossian formulation of the Kashiwara-Vergne equations, and most recently, a topological description of the Kashiwara-Vergne groups. The “recent” portion of the talk is based on joint work with Iva Halacheva and Marcy Robertson (arXiv: 2106.02373), and joint work with Dror Bar-Natan (arXiv: 1405.1955).

Classification of string links up to 2n-moves and link-homotopy

May 31, 2021May 30, 2021 | chrisman.76

Kodai Wada (Kobe University)

$2n$

Arrow calculus for welded links

April 19, 2021March 31, 2021 | chrisman.76

Akira Yasuhara (Waseda University)

We develop a calculus for diagrams of knotted objects. We define arrow presentations, which encode the crossing information of a diagram into arrows in a way somewhat similar to Gauss diagrams, and more generally w–tree presentations, which can be seen as “higher-order Gauss diagrams”. This arrow calculus is used to develop an analogue of Habiro’s clasper theory for welded knotted objects, which contain classical link diagrams as a subset. This provides a “realization” of Polyak’s algebra of arrow diagrams at the welded level, and leads to a characterization of finite- type invariants of welded knots and long knots. This is a joint work with Jean-Baptiste Meilhan (University of Grenoble Alpes).

Characterization(s) of the Reduced Peripheral System

March 22, 2021March 22, 2021 | chrisman.76

Jean-Baptiste Meilhan (Université Grenoble Alpes)

The reduced peripheral system was introduced by Milnor in the 50’s for the study of links up to link-homotopy, i.e. up to homotopies leaving distinct components disjoint. This invariant, however, fails to classify links up to link-homotopy for links of 4 or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. We give a topological characterization in terms of ribbon solid tori in 4-space up to link-homotopy, using a version of Artin’s Spun construction. The proof relies heavily on an intermediate characterization, in terms of welded links up to self-virtualization, providing hence a purely topological application of the combinatorial theory of welded links.

Cut-diagrams and a Chen–Milnor presentation result for knotted surfaces

February 22, 2021February 19, 2021 | chrisman.76

Benjamin Audoux (Aix-Marseille Université)

In this talk, I will introduce cut-diagrams, a combinatorial data which generalizes welded links to higher dimensions. Using them, I will give and discuss then Chen-Milnor presentations for the nilpotent and reduced fundamental groups of knotted surfaces. This is joint work in progress with J-B. Meilhan and A. Yasuhara.

Extending Milnor’s concordance invariants to virtual knots and welded links

May 18, 2020May 18, 2020 | chrisman.76

Micah Chrisman (Ohio State University)

Milnor’s $\bar{\mu}$ -invariants for links in the 3-sphere vanish on any link concordant to a boundary link. In particular, they are are trivial for any classical knot. Here we define an analogue of Milnor’s concordance invariants for knots in thickened surfaces $\Sigma \times [0,1]$ , where $\Sigma$ is closed and oriented. These invariants vanish on any knot concordant to a homologically trivial knot in $\Sigma \times [0,1]$ . We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index polynomial, and generalized Alexander polynomial. Moreover, we complete the slice status classification of the 2564 virtual knots having at most five classical crossings and reduce to four (of 90235) the number of virtual knots with six classical crossings having unknown slice status. Furthermore, we prove that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian. As part of the construction of the extended $\bar{\mu}$ -invariants, we also obtain a generalization of the $\bar{\mu}$ -invariants of classical links in $S^3$ to ribbon torus links in $S^4$ and welded links.