# Posts

## What is a braidoid diagram?

### Neslihan Gügümcü (Izmir Institute of Technology and University of Goettingen)

In this talk we first review the basics of the theory of knotoids introduced by Vladimir Turaev in 2012 [1]. A knotoid diagram is basically an open-ended knot diagram with two open endpoints that can lie in any local region complementary to the plane of the diagram. The theory of knotoids extends the classical knot theory and brings up some interesting problems and features such as the height problem [1,3] and parity notion and related invariants such as off writhe and parity bracket polynomial [4]. It was a curious problem to determine a “braid like object” corresponding to knotoid diagrams. The second part of this talk is devoted to the theory of braidoids, introduced by the author and Sofia Lambropoulou [2]. We present the notion of a braidoid and analogous theorems to the classical Alexander Theorem and the Markov Theorem, that relate knotoids/multi-knotoids in the plane to braidoids.

[1] V.Turaev, Knotoids, Osaka J.of Mathematics 49 (2012), 195–223. https://arxiv.org/abs/1002.4133

[2] N.Gugumcu and S.Lambropoulou, Braidoids, to appear in Israel J.of Mathematics, https://arxiv.org/abs/1908.06053

[3] N.Gugumcu and L. Kauffman, New Invariants of Knotoids, European J.of Combinatorics, (2017), 65C, 186-229, https://arxiv.org/abs/1602.03579

[4] The Guassian parity and minimal diagrams of knot-type knotoids, submitted.

## Invariants of framed links from cohomology of ternary self-distributive structures

### Emanuele Zappala (University of South Florida)

In this talk I recall the definitions of shelf/rack/quandle and their cohomology theory. I also give the construction of cocycle invariant of links, due to Carter, Jelsovsky, Kamada, Langford and Saito (Trans. Amer. Math. Soc. 355 (2003), 3947-3989). Then, I introduce higher arity self-distributive structures and show that an appropriate diagrammatic interpretation of them is suitable to define a ternary version of the cocycle invariant for framed link invariants, via blackboard framings. I discuss the computation of cohomology of ternary structures, as composition of mutually distributive operations, and a cohomology theory of certain ternary quandles called group heaps. Furthermore I mention a categorical version of self-distributivity, along with examples from Lie algebras and Hopf algebras, and the construction of ribbon categories from ternary operations that provide a quantum interpretation of the (ternary) cocycle invariant.

### References:

Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Higher Arity Self-Distributive Operations in Cascades and their Cohomology.” arXiv preprint arXiv:1905.00440 (2019).

Elhamdadi, Mohamed, Masahico Saito, and Emanuele Zappala. “Heap and Ternary Self-Distributive Cohomology.” arXiv preprint arXiv:1910.02877 (2019).

Zappala, Emanuele. “Non-Associative Algebraic Structures in Knot Theory.” (2020).

## Lower bounds on the tunnel number of composite spatial theta graphs

### Scott Taylor (Colby College)

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots.The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs,using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

### Lou Kauffman (University of Illinois-Chicago & Novosibirsk State University)

This talk discusses virtual knot theory and rotational virtual knot theory. In virtual knot theory we introduce a virtual crossing in the diagrams along with over crossings and under crossings. Virtual crossings are artifacts of representing knots in higher genus surfaces as diagrams in the plane.

Virtual diagrammatic equivalence is the same as studying knots in thickened surfaces up to ambient isotopy, surface homeomorphisms and handle stabilization. At the diagrammatic level, virtual knot equivalence is generated by Reidemeister moves plus detour moves. In rotational virtual knot theory, the detour moves are restricted to regular homotopy of plane curves (the self-crossings are virtual). Rotational virtual knot theory has the property that all classical quantum link invariants extend to quantum invariants of rotational virtual knots and links. We explain this extension, and we consider the question of the power of quantum invariants in this context.

By considering first the bracket polynomial and its extension to a parity bracket polynomial for virtual knots (Manturov) and its further extension to a rotational parity bracket polynomial for knots and links (Kaestner and Kauffman), we give examples of links that are detected via the parity invariants that are not detectable by quantum invariants. In the course of the discussion we explain a functor from the rotational tangle category to the diagrammatic category of a quantum algebra. We delineate significant weaknesses in quantum invariants and how these gaps can be fulfilled by taking parity into account.

The slides for the talk are given below:

RotationalVirtualKnotsKeynote.key

### References

Kauffman, Louis H, Rotational virtual knots and quantum link invariants, https://arxiv.org/abs/1509.00578

## 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

## Hyperbolicity and Turaev hyperbolicity of virtual knots

Abstract: Hyperbolic volume has been a powerful invariant for classical knots. In this talk we extend it to virtual knots, including calculations of virtual knot volumes. We further introduce Turaev volume for knots and virtual knots and show that EVERY knot has a Turaev volume.

References:

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Turaev Hyperbolicity of Classical and Virtual Knots, https://arxiv.org/pdf/1912.09435.pdf

Adams, C., Eiseberg, E., Greenberg, J. Kapoor, K., Liang, Z., O’Connor, K., Pacheco-Tallaj, N., Wang, Y., Tg-Hyperbolicity of Virtual Links, https://arxiv.org/pdf/1904.06385.pdf

## Maggie Miller (Princeton University)

Abstract: David Gabai showed that a minimum-genus surface in a knot complement remains minimum-genus when capped off into the zero-surgery on that knot, implying that surgery on a nontrivial knot can never yield $S^1\times S^2$ (thus proving the Property R conjecture). I study this problem for links. In particular, I show that if $L$ is a 2-component link with nonzero linking number and nondegenerate Thurston norm on its complement, then there exists a finite set $E\subset H_2(S^3\setminus\nu(L),\partial;\mathbb{R})$ so that if $S$ is norm-minimizing and not in $E$ up to scalar multiplication, then $\hat{S}$ is norm-minimizing in the 3-manifold obtained from $S^3$ by doing Dehn surgery on $L$ according to $\partial S$. (The result generally holds for $n>1$-component links with $E$ $(n-2)$-dimensional.) The proof involves constructing a taut foliation on $S^3\setminus\nu(L)$ with nice boundary properties, motivated by Gabai’s proof of the Property R conjecture.

References:

Miller, M., The effect of link Dehn surgery on the Thurston norm. https://arxiv.org/abs/1906.08458