## Diagrammatic Algebra, Part 1

### J. Scott Carter (University of South Alabama)

This talk is based upon joint work with Seiichi Kamada. Abstract tensor notation for multi-linear maps uses boxes with in-coming and out-going strings to represent structure constants. A judicious choice of variables for such constants leads to the diagrammatic representation: boxes are replaced by glyphs. One of the most simple examples of a multi-category is the diagrammatic representation of embedded surfaces in 3-space that we all grew up learning. The standard drawing of a torus (an oval, a smile and a moustache) is a representation based upon drawings of surface singularities. We start from a two object category with a pair of arrows that relate them. Cups and caps are constructed easily. From these, births, deaths, saddles, forks, and cusps are created as triple arrows. At the top level, there are critical cancellations, lips, beak-to-beak, horizontal cusps, and swallow-tails. Interestingly, the structure extends inductively to describe many relations about handles in higher dimensions.

CKVK* Recommendation: The talk has many beautiful figures that are best viewed on Youtube in HD.

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

### 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.

Slides for the talk are available here.

#### References:

Boden, H.U., Chrisman, M., “Virtual concordance and the generalized Alexander polynomial”, https://arxiv.org/pdf/1903.08737.pdf

Chrisman, M., “Milnor’s concordance invariants for knots on surfaces”, https://arxiv.org/pdf/2002.01505.pdf

## Knots with critical bridge spheres

### Puttipong Pongtanapaisan (University of Iowa)

David Bachman introduced the notion of critical surfaces and showed that they satisfy useful properties. In particular, they behave like incompressible surfaces and strongly irreducible surfaces. In this talk, I will review some techniques that have been used to study bridge spheres and give examples of nontrivial knots with critical bridge spheres. This is joint work with Daniel Rodman.

References:

Pongatanpaisan, P., Rodman, D., Critical Bridge Spheres for Links with Arbitrarily Many Bridges, https://arxiv.org/pdf/1906.03483.pdf