Homotopy type of circle graph complexes motivated by extreme Khovanov homology.

*(English)*Zbl 1431.57012The authors work along the lines of González-Meneses, Manchón and Silvero [J. González-Meneses et al., Proc. R. Soc. Edinb., Sect. A, Math. 148, No. 3, 541–557 (2018; Zbl 1428.57006)] who showed that the extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex obtained from a bipartite circle graph constructed from the diagram. The authors conjecture that this simplicial complex is always homotopy equivalent to a wedge of spheres. In particular, its homotopy type, if not contractible, would be a link invariant (up to suspension), and it would imply that the extreme Khovanov homology of any link diagram does not contain torsion:

Conjecture

Conversely, the authors give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres and they also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem shedding light on previous results by P. Csorba [Electron. J. Comb. 16, No. 2, Research Paper R11, 7 p. (2009; Zbl 1171.05384)], U. Nagel and V. Reiner [ibid. 16, No. 2, Research Paper R3, 59 p. (2009; Zbl 1186.13022)], and J. Jonsson [Simplicial complexes of graphs. Berlin: Springer (2008; Zbl 1152.05001)]. Finally they study the implications of their results to knot theory; more precisely, they compute the real-extreme Khovanov homology of torus links \(T(3,q)\) and obtain examples of \(H\)-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.

In Section 2 they review basic definitions and present the conjecture that they will deal with throughout the work like wedges, joins and independence simplicial complexes. This work is motivated by the study of extreme Khovanov homology of link diagrams to coincide with the reduced cohomology of the independence simplicial complex associated with the Lando graph constructed from the link diagram. This is the reason for considering those particularizations of the general conjecture.

In Section 3 the authors describe the classical idea of building the simplicial complex cone by cone (in essence the cell decomposition of the complex) and present the first results involving the families of cactus and outerplanar graphs. They introduce some results that will be useful throughout the paper. They start by reviewing the homotopy type of the independence complexes of paths, trees, and cycles, and they prove the Conjecture for the families of cactus and outerplanar graphs. In order to simplify notation, they delete \(\mathcal{K}\) from \(\text{lk}_{\mathcal{K}}(v)\) and \(\text{st}_{\mathcal{K}}(v)\) when it is clear from context which simplicial complex is considered.

In Sections 4 and 5 they prove their conjecture for the family of permutation graphs and non-nested circle graphs, respectively. They prove also Conjecture 2.11 for the family of non-nested circle graphs, which are bipartite. The study of bipartite circle graphs is relevant since these are the graphs arising as Lando graphs associated to link diagrams.

Section 6 is devoted to proving a general theorem on independence complexes which sheds light on some results by Csorba [Zbl 1171.05384], and Nagel and Reiner [Zbl 1186.13022]. The authors actually apply the principle for gluing homotopies to obtain several useful properties of graphs and their independence complexes. In particular, Theorem 6.4 allows them to generalize Theorem 3.10 by Csorba [Zbl 1171.05384], the bipartite suspension theorem by Nagel and Reiner [Zbl 1186.13022] and its generalization by Jonsson [Zbl 1152.05001]. They start from a series of simple but useful lemmas. Given a vertex \(v\) of a loopless graph \(G\), write \(A_v=I_{G-st(v)}*v\). The authors also use the standard notation \(N_G(v)\) for the set of neighbors of the vertex \(v\) in the graph \(G\). Note that \(N_G(v)\) is the set of vertices of \(lk_G(v)\).

Finally, in Section 7 they show some applications of their work to knot theory, namely they compute the extreme Khovanov homology of torus links \(T(3,q)\) and construct two families of \(H\)-thick knots having two and three non-trivial extreme Khovanov homology groups separated by gaps as long as desired. Khovanov homology is a powerful link invariant introduced by M. Khovanov [Duke Math. J. 101, No. 3, 359–426 (2000; Zbl 0960.57005)] in the end of the last century; the authors start to review this well-known theory. O. Viro in [Fundam. Math. 184, 317–342 (2004; Zbl 1078.57013)] introduced a nice and important view of Khovanov cohomology by using Kauffman’s technique of enhanced states which in my opinion gives another missing part of the Khovanov story for link homology. The present authors present an overview of the history and ideas of enhanced states and then pass on to the Lando graph of a link diagram that they use later according to the relations to the Jones polynomial and hence Khovanov cohomology. This fact comes from the independence number. The Jones polynomial can be seen as the Euler characteristic of Khovanov cohomology and the independence number also suggests the formula of Euler characteristic. Section 7 discusses extreme Khovanov homology as the homology of the independence complex of bipartite circle graphs, extreme Khovanov homology of torus links \(T(3,q)\), and gaps in extreme Khovanov homology.

This paper presents good results and gives new ideas and methods useful for researchers and PhD students who are interested in studying knot theory and Khovanov homology and its applications.

Conjecture

- (1)
- The independence complex associated with a circle graph is homotopy equivalent to a wedge of spheres.
- (2)
- In particular, the independence complex associated with a bipartite circle graph (Lando graph) is homotopy equivalent to a wedge of spheres.
- (3)
- As a consequence, the extreme Khovanov homology of any link diagram is torsion-free.

Conversely, the authors give a method for constructing a permutation graph whose independence simplicial complex is homotopy equivalent to any given finite wedge of spheres and they also present some combinatorial results on the homotopy type of finite simplicial complexes and a theorem shedding light on previous results by P. Csorba [Electron. J. Comb. 16, No. 2, Research Paper R11, 7 p. (2009; Zbl 1171.05384)], U. Nagel and V. Reiner [ibid. 16, No. 2, Research Paper R3, 59 p. (2009; Zbl 1186.13022)], and J. Jonsson [Simplicial complexes of graphs. Berlin: Springer (2008; Zbl 1152.05001)]. Finally they study the implications of their results to knot theory; more precisely, they compute the real-extreme Khovanov homology of torus links \(T(3,q)\) and obtain examples of \(H\)-thick knots whose extreme Khovanov homology groups are separated either by one or two gaps as long as desired.

In Section 2 they review basic definitions and present the conjecture that they will deal with throughout the work like wedges, joins and independence simplicial complexes. This work is motivated by the study of extreme Khovanov homology of link diagrams to coincide with the reduced cohomology of the independence simplicial complex associated with the Lando graph constructed from the link diagram. This is the reason for considering those particularizations of the general conjecture.

In Section 3 the authors describe the classical idea of building the simplicial complex cone by cone (in essence the cell decomposition of the complex) and present the first results involving the families of cactus and outerplanar graphs. They introduce some results that will be useful throughout the paper. They start by reviewing the homotopy type of the independence complexes of paths, trees, and cycles, and they prove the Conjecture for the families of cactus and outerplanar graphs. In order to simplify notation, they delete \(\mathcal{K}\) from \(\text{lk}_{\mathcal{K}}(v)\) and \(\text{st}_{\mathcal{K}}(v)\) when it is clear from context which simplicial complex is considered.

In Sections 4 and 5 they prove their conjecture for the family of permutation graphs and non-nested circle graphs, respectively. They prove also Conjecture 2.11 for the family of non-nested circle graphs, which are bipartite. The study of bipartite circle graphs is relevant since these are the graphs arising as Lando graphs associated to link diagrams.

Section 6 is devoted to proving a general theorem on independence complexes which sheds light on some results by Csorba [Zbl 1171.05384], and Nagel and Reiner [Zbl 1186.13022]. The authors actually apply the principle for gluing homotopies to obtain several useful properties of graphs and their independence complexes. In particular, Theorem 6.4 allows them to generalize Theorem 3.10 by Csorba [Zbl 1171.05384], the bipartite suspension theorem by Nagel and Reiner [Zbl 1186.13022] and its generalization by Jonsson [Zbl 1152.05001]. They start from a series of simple but useful lemmas. Given a vertex \(v\) of a loopless graph \(G\), write \(A_v=I_{G-st(v)}*v\). The authors also use the standard notation \(N_G(v)\) for the set of neighbors of the vertex \(v\) in the graph \(G\). Note that \(N_G(v)\) is the set of vertices of \(lk_G(v)\).

Finally, in Section 7 they show some applications of their work to knot theory, namely they compute the extreme Khovanov homology of torus links \(T(3,q)\) and construct two families of \(H\)-thick knots having two and three non-trivial extreme Khovanov homology groups separated by gaps as long as desired. Khovanov homology is a powerful link invariant introduced by M. Khovanov [Duke Math. J. 101, No. 3, 359–426 (2000; Zbl 0960.57005)] in the end of the last century; the authors start to review this well-known theory. O. Viro in [Fundam. Math. 184, 317–342 (2004; Zbl 1078.57013)] introduced a nice and important view of Khovanov cohomology by using Kauffman’s technique of enhanced states which in my opinion gives another missing part of the Khovanov story for link homology. The present authors present an overview of the history and ideas of enhanced states and then pass on to the Lando graph of a link diagram that they use later according to the relations to the Jones polynomial and hence Khovanov cohomology. This fact comes from the independence number. The Jones polynomial can be seen as the Euler characteristic of Khovanov cohomology and the independence number also suggests the formula of Euler characteristic. Section 7 discusses extreme Khovanov homology as the homology of the independence complex of bipartite circle graphs, extreme Khovanov homology of torus links \(T(3,q)\), and gaps in extreme Khovanov homology.

This paper presents good results and gives new ideas and methods useful for researchers and PhD students who are interested in studying knot theory and Khovanov homology and its applications.

Reviewer: Ahmad Al Yasry (Bonn)

##### MSC:

57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |

57M15 | Relations of low-dimensional topology with graph theory |

05E45 | Combinatorial aspects of simplicial complexes |

##### Keywords:

circle graphs; independence simplicial complex; Khovanov homology; torus links; wedge of spheres
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\textit{J. H. Przytycki} and \textit{M. Silvero}, J. Algebr. Comb. 48, No. 1, 119--156 (2018; Zbl 1431.57012)

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