Turaev Genus Research Articles

Near extremal Khovanov homology of Turaev genus one links

The Turaev surface of a link diagram D is a closed, oriented surface constructed from a cobordism between the all-A and all-B Kauffman states of D. The Turaev genus of a link L is the minimum genus of the Turaev surface of any diagram D of L. A link is alternating if and only if its Turaev genus is zero, and so one can view Turaev genus one links as being close to alternating links. In this paper, we study the Khovanov homology of a Turaev genus one link in the first and last two polynomial gradings where the homology is nontrivial. We show that a particular summand in the Khovanov homology of a Turaev genus one link is trivial. This trivial summand leads to a computation of the Rasmussen s invariant and to bounds on the smooth four genus for certain Turaev genus one knots.

Concordance Invariants and the Turaev Genus

Abstract We show that the differences between various concordance invariants of knots, including Rasmussen’s $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the 1st example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar–Kofman.

The Khovanov Homology of Alternating Virtual Links

In this paper, we study the Khovanov homology of an alternating virtual link L and show that it is supported on g+2 diagonal lines, where g equals the virtual genus of L. Specifically, we show that Khi,j(L) is supported on the lines j=2i−σξ+2k−1 for 0≤k≤g+1 where σξ∗(L)+2g=σξ(L) are the signatures of L for a checkerboard coloring ξ and its dual ξ∗. Of course, for classical links, the two signatures are equal, and this recovers Lee’s H-thinness result for Kh∗,∗(L). Our result applies more generally to give an upper bound for the homological width of the Khovanov homology of any checkerboard virtual link L. The bound is given in terms of the alternating genus of L, which can be viewed as the virtual analogue of the Turaev genus. The proof rests on associating, with any checkerboard colorable link L, an alternating virtual link diagram with the same Khovanov homology as L. In the process, we study the behavior of the signature invariants under vertical and horizontal mirror symmetry. We also compute the Khovanov homology and Rasmussen invariants in numerous cases and apply them to show nonsliceness and determine the slice genus for several virtual knots. ??Table]ras-table at the end of the paper lists the signatures, Khovanov polynomial, and Rasmussen invariant for alternating virtual knots up to six crossings.

Extremal Khovanov homology of Turaev genus one links

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.

The Turaev genus of torus knots

The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at most one. We also determine the dealternating number of a torus knot with five or fewer strands up to an error of at most two. We also give bounds on the Turaev genus and dealternating number of torus links with five or fewer strands and on some infinite families of torus links on six strands.

Link diagrams with low Turaev genus

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating.

Infinite families of hyperbolic prime knots with alternation number 1 and dealternating number n

For each positive integer [Formula: see text] we will construct a family of infinitely many hyperbolic prime knots with alternation number 1, dealternating number equal to [Formula: see text], braid index equal to [Formula: see text] and Turaev genus equal to [Formula: see text].

Turaev genus and alternating decompositions

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.

The graded count of quasi-trees is not a knot invariant

In "A survey on the Turaev genus of knots," Champanerkar and Kofman propose several open questions. The first asks whether the polynomial whose coefficients count the number of quasi-trees of the all-A ribbon graph obtained from a diagram with minimal Turaev genus is an invariant of the knot. We answer negatively by showing a counterexample obtained from the two diagrams of $8_{21}$ on the KnotAtlas and KnotScape.

On matrix-model approach to simplified Khovanov–Rozansky calculus

Wilson-loop averages in Chern–Simons theory (HOMFLY polynomials) can be evaluated in different ways – the most difficult, but most interesting of them is the hypercube calculus, the only one applicable to virtual knots and used also for categorification (higher-dimensional extension) of the theory. We continue the study of quantum dimensions, associated with hypercube vertices, in the drastically simplified version of this approach to knot polynomials. At q=1 the problem is reformulated in terms of fat (ribbon) graphs, where Seifert cycles play the role of vertices. Ward identities in associated matrix model provide a set of recursions between classical dimensions. For q≠1 most of these relations are broken (i.e. deformed in a still uncontrollable way), and only few are protected by Reidemeister invariance of Chern–Simons theory. Still they are helpful for systematic evaluation of entire series of quantum dimensions, including negative ones, which are relevant for virtual link diagrams. To illustrate the effectiveness of developed formalism we derive explicit expressions for the 2-cabled HOMFLY of virtual trefoil and virtual 3.2 knot, which involve respectively 12 and 14 intersections – far beyond any dreams with alternative methods. As a more conceptual application, we describe a relation between the genus of fat graph and Turaev genus of original link diagram, which is currently the most effective tool for the search of thin knots.

A Survey on the Turaev Genus of Knots

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.

Turaev genus, knot signature, and the knot homology concordance invariants

We give bounds on knot signature, the Ozsvath-Szabo T invariant, and the Kasmussen s invariant in terms of the Turaev genus ot the knot.

The Turaev genus of an adequate knot

The Turaev genus of a knot is an obstruction to the knot being alternating. An adequate knot is a generalization of an alternating knot. A natural problem is a characterization of the Turaev genus of an adequate knot. In this paper, we show that the Turaev genus of an adequate knot is realized by the genus of the Turaev surface associated to an adequate diagram of the knot using the Khovanov homology. As a result, we obtain the additivity of the Turaev genus of adequate knots, and show that the Turaev genus of an adequate knot is “often” preserved under mutation. We also show that an n-semi-alternating knot is of Turaev genus n. This is the first examples of adequate knots of Turaev genus two or more.

On knot Floer width and Turaev genus

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizontal distance between two such lines. Also, for each diagram $D$ of $K$ there is an associated Turaev surface, and the Turaev genus is the minimum genus of all Turaev surfaces for $K$. We show that the width of knot Floer homology is bounded by Turaev genus plus one. Skein relations for genus of the Turaev surface and width of a complex that generates knot Floer homology are given.

Invariants for Turaev genus one links

The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.