Sharpness of the Morton-Franks-Williams inequality
Ilya Alekseev (Euler International Mathematical Institute)

We aim to study the braid index, which is a natural complexity measure of knots and links. In practice, to estimate the braid index, one can apply the Morton-Franks-Williams inequality. This inequality relates the breadth of the HOMFLY-PT polynomial to the number of Seifert circles. For alternating links, Y. Diao, G. Hetyei, and P. Liu found a simple geometric criterion for the sharpness of the Morton-Franks-Williams inequality. In the present talk, we extend their results to larger classes of links.

Knot invariants with multiple skein relations containing virtual crossings
Danish Ali (Dalian University of Technology)

Knot invariants for oriented knot diagrams can be constructed using skein relations. Traditionally, each knot invariant has only one skein relation, and the skein relation has three or four terms. In this paper, we use a system of skein equations to construct knot invariants, and each skein equation has six or eight terms. The multiple skein relations use new ways to smooth or virtualize a crossing. This invariant can also be altered by writhe to get a more powerful invariant. Today there are many knot invariants, but still, they can not distinguish all classical knots. One motivation of this research is to find a more powerful invariant. The definition of this invariant uses virtual crossings. We gave an example that both Kauffman two-variable polynomial and HOMFLY polynomials fail to distinguish a pair of knots, but a simplified version of this invariant can easily distinguish those knots. Virtual crossings play a key role in this construction. If the virtual terms are deleted, the simplified version invariant become trivial.

Algorithm for constructing a rectangular diagram of the Seifert surface
Mikhail Chernavskikh (Moscow State University)

Rectangular diagrams of links are widely used in knot theory. Dynnikov-Prasolov introduced rectangular diagrams of surfaces. We present an algorithm for constructing a rectangular diagram of the Seifert surface for any link, represented by a rectangular diagram. We estimate the complexity of a resulting diagram of a surface.

Symplectic Khovanov homology and Jones polynomial
Zhechi Cheng (Wuhan University)

Symplectic Khovanov homology was defined to be a singly graded link invariant that is conjecturally isomorphic to Khovanov homology with its two gradings collapsed. Over characteristic 0 fields, the conjecture was proved by Abouzaid and Smith. In this talk, I generalize symplectic Khovanov homology to a doubly graded invariant and explain how to recover the Jones grading on symplectic Khoavnov homology through generalizing Abouzaid-Smith isomorphism to a bigraded version.

On checkerboard colorability and arrow polynomial
Qingying Deng (Xiangtan University)

It is well-known that a classical link diagram is checkerboard colorable. The notion of a checkerboard coloring for a virtual link diagram was independently introduced by V.O. Manturov (in 2000) and N. Kamada (in 2002) by using atom and corresponding abstract link diagram, respectively. M.O. Bourgoin introduced the twisted knot theory in 2008 and defined the notion of a checkerboard coloring for a twisted link diagram. In this talk, we first give two new criteria to detect the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. Then by applying these criteria we determine the checkerboard colorability of virtual knots up to four crossings, with only one exception. Second, we reformulate the arrow polynomial of twisted links by using Kauffman’s items. In fact, in 2012, in case of using the pole diagram, N. Kamada obtained the polynomial by generalizing a multivariable polynomial invariant of a virtual link to a twisted link. Moreover, we figure out three characteristics of the arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.

On volumes of hyperbolic right-angled polyhedra
Andrei Egorov (Novosibirsk State University)

In three-dimensional hyperbolic space consider polyhedra with all dihedral angles equal to pi/2. We will look at some properties of this polyhedra and consider new upper bounds on hyperbolic volumes this polyhedra in three independent cases: for ideal polyhedra with all vertices on the boundary of hyperbolic space, for compact polyhedra with only finite vertices, and for polyhedra with vertices of both types. In addition, we will look at some connections with knot theory and fullerenes.

Sequence of virtual link invariants arising from flat links
Maxim Ivanov (Novosibirsk State University)

Flat virtual links are equivalence classes of virtual links with respect to a changing of a type of a classical crossing in a diagram. We present a recurrent construction of invariants of virtual link by using invariants of flat virtual links. Those invariants appear to be useful in studying connected sums of virtual knots. As an example, we give a new proof of Kishino knot being nontrivial knot.

The singular braid group
Tatyana Kozlovskaya (Tomsk State University)

We study the singular braid group and the singular pure braid group. We find generators, defining relations and the algebraical structure of the singular pure braid group. Also we construct linear representations and representation by automorphisms of free group for the singular braid group.

Several faces of cobordisms: matchings, pictures, and groups
Vassily Manturov (Moscow Institute of Physics and Technology)

My talk will be devoted to sliceness obstructions for various analogues of knots. We shall be mostly concerned with the two cases:
1. Free knots (capped by formal folded 2-discs).
2. Knots in the full torus (to be capped by a disc in S¹×D³).
A naive approach requires some "matchings" between crossings to be paired: say, for a 2-component link a mixed crossing can not be paired with a pure crossing. We'll see that one can strongly generalise the "matching" approach. One of the main results says that (under some very mild conditions) if the knot is slice then it is elementarily slice (can be capped by a disc with only double point singularities). The second part of the talk will be devoted to slineness obstructions valued in groups which are (commensurable to) free product and direct products of cyclic groups.

Gordian graphs
Alexey Miller (Euler International Mathematical Institute)

Often in knot theory problems we can see such geometric procedures that change the isotopic type of a knot. The most popular type of such procedure is a local transformation (or local move). A local move on a link is the substitution of a given tangle for another. A local move always transforms a link into a link, possibly the same one. The most famous local transformations are X-move (also called crossing change), band surgery, #-move, Delta-move, Clasp-pass-move and many more. A Gordian graph is a versatile tool for working with local transformations. This talk will present both well-known results on local transformations, reformulated in terms of Gordian graphs, and our new results obtained with this useful tool (behavior of a Gordian graph at infinity, partial description of a unit neighborhood of an arbitrary vertex of a Gordian graph).

Intersection formulas for parities on virtual knots
Igor Nikonov (Moscow State University)

We show that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows to define a new series of parities on virtual knots.

Invariants of knots in a solid torus
Semen Panenko (Moscow Institute of Physics and Technology)

I will talk about the constructed invariant for a knot in a solid torus, based on the use of homology classes of loops, and also about its application to the proof of the nontriviality of knots.

Minimal length of nonsimple closed geodesics on hyperbolic surfaces
Wujie Shen (Peking University)

The length of a simple closed geodesic on a hyperbolic surface can be arbitrarily small, and Hempel showed that a nonsimple closed geodesic has a universal lower bound 2 log(1+√2). We will give some estimates on the self-intersection number and length of a nonsimple geodesic on a hyperbolic surface.

Hyperellipticity of closed orientable Euclidean 3-manifolds
Bao Vuong (Tomsk State University and Novosibirsk State University)

We study closed orientable Euclidean 3-manifolds which are also known as flat 3-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, R.H. Fox showed that the 3-torus is not a double branched covering of the 3-sphere. Then, it is not a hyperelliptic manifold. We show that all the remaining Euclidean 3-forms are hyperelliptic manifolds. This is a joint work with A.D. Mednykh.

The relation between tunnel numbers of a cable knot and its companion
Junhua Wang (Xi'an Jiaotong-Liverpool University)

In this talk, we consider tunnel numbers of cable knots. In previous work with Yanqing Zou, we proved the tunnel number of a cable knot is not less than that of its companion, but not more than that of its companion plus one. Here we try to give a sufficient and necessary condition of when they are equal, especially for tunnel number one knots. It seems that there is some relation to doubly primitive in Berge's conjecture. This is a joint work with Jiajun Wang.

Chord index for knots in thickened surfaces
Mengjian Xu (Guangxi Normal University)

Inspired by the works of Boden and Rushworth, we construct a homomorphism from a subgroup of the first homology group of the surface to the group of chord indices (will be defined) of a knot in a thickend surface. From this construction, we will obtain many invariants of knots in thicken surfaces. And we apply these invariants to virtual knot theory. We will use some examples to explain our theory.

Heegaard Floer d-invariant and band surgeries on the torus knot T(2,n)
Jingling Yang (The Chinese University of Hong Kong)

Lifting to double branched covers, a band surgery problem on knots becomes a Dehn surgery problem between lens spaces, which are the so-called L-space in Heegaard Floer theory. In this talk, we deduce a Heegard Fleor d-invariant surgery formula for a general L-space, and we apply the formula to give an almost complete classification of band surgeries on torus knot T(2,n). Precisely, we show that the torus link T(2,s) for s ≠ 0 is obtained by a band surgery from T(2,n) with n ≥ 5 odd only if n and s satisfy one of the following cases:
(1) n ≥ 5 is any odd integer and s = ± 1, n, n ± 1 or n ± 4;
(2) n = 5 and s = −5;
(3) n = 5 and s = −9;
(4) n = 9 and s = −5.
As a corollary, the only nontrivial torus knot T(2,n) admitting chirally cosmetic banding is T(2,5).

Higher-dimensional Heegaard Floer homology and Hecke algebras
Tianyu Yuan (University of California, Los Angeles)

The higher-dimensional Heegaard Floer homology (HDHF) of a symplectic manifold is a model for the Floer homology on its Hilbert scheme. Given a closed oriented surface of genus greater than zero, we construct a map from the HDHF of the cotangent fibers to the Hecke algebra associated to the surface and show that it is an isomorphism of algebras. For example, the surface Hecke algebra of a 2-torus is a double affine Hecke algebra. This is joint work with Ko Honda and Yin Tian.

Algebraic fibrations of 4-manifolds
that cover certain right-angled hyperbolic 4-polytopes
Fangting Zheng (Xi'an Jiaotong-Liverpool University)

Algebraic fibration is a generalization of the fibered 3-manifold in higher dimensions. In this talk, we will discuss hyperbolic 4-manifolds M of small volume that fiber algebraically. This is a joint work with Jiming Ma.