Turaev Surfaces

TL;DR

This paper provides an expository overview of Turaev surfaces, explaining their mathematical properties and significance in knot theory for an encyclopedia entry.

Contribution

It offers a clear, accessible explanation of Turaev surfaces, highlighting their role and importance in the study of knots.

Findings

Clarifies the construction of Turaev surfaces

Explains their application in knot invariants

Provides foundational knowledge for further research

Abstract

An expository article on Turaev surfaces written for "A Concise Encyclopedia of Knot Theory," to appear.

Figures

Equations

$$g_T(D)=g(F(D))=(c(D)+2-|s_A|-|s_B|)/2.$$

$$\mathrm{span}{V}_L(t)\le c(L)-g_T(L)$$

$$g_T(L)=g_T(D)=\frac{1}{2}\left(c(D)-|s_A(D)|-|s_B(D)|\right)+1=c(L)-\mathrm{span}{V}_L(t).$$

$$w_{KH}(K)-2\le g_T(K)\text{and}{w}_{HF}(K)-1\le g_T(K).$$

Full text

Turaev Surfaces

Seungwon Kim

National Institute for Mathematical Sciences, Daejeon, South Korea

 and 

Ilya Kofman

Department of Mathematics, College of Staten Island & The Graduate Center, City University of New York, New York, USA

An expository article written for A Concise Encyclopedia of Knot Theory, to appear.

1. Introduction

The two most famous knot invariants, the Alexander polynomial (1923) and the Jones polynomial (1984), mark paradigm shifts in knot theory. After each polynomial was discovered, associating new structures to knot diagrams played a key role in understanding its basic properties. Using the Seifert surface, Seifert showed, for example, how to obtain knots with a given Alexander polynomial. For the Jones polynomial, even the simplest version of that problem remains open: Does there exist a non-trivial knot with trivial Jones polynomial?

Kauffman gave a state sum for the Jones polynomial, with terms for each of the $2^{c}$ states of a link diagram with $c$ crossings. Turaev constructed a closed orientable surface from any pair of dual states with opposite markers.

For the Jones polynomial, the Turaev surface is a rough analog to the Seifert surface for the Alexander polynomial. For a given knot diagram, the Seifert genus and the Turaev genus are computed by separate algorithms to obtain each surface from the diagram. The invariants for a given knot $K$ are defined as the minimum genera among all the respective surfaces for $K$. The Seifert genus is a topological measure of how far a given knot is from being unknotted. The Turaev genus is a topological measure of how far a given knot is from being alternating. (See [26], which discusses alternating distances.) For any alternating diagram, Seifert’s algorithm produces the minimal genus Seifert surface. For any adequate diagram, Turaev’s algorithm produces the minimal genus Turaev surface. Extending the analogy, we can determine the Alexander polynomial and the Jones polynomial of $K$ from associated algebraic structures on the respective surfaces of $K$: the Seifert matrix for the Alexander polynomial, and the $A$–ribbon graph on the Turaev surface for the Jones polynomial.

The analogy is historical, as well. Like the Seifert surface for the Alexander polynomial, the Turaev surface was constructed to prove a fundamental conjecture related to the Jones polynomial. In the 1880’s, Tait conjectured that an alternating link always has an alternating diagram that has minimal crossing number among all diagrams for that link. A proof had to wait about a century until the Jones polynomial led to several new ideas used to prove Tait’s Conjecture [21, 29, 35]. Turaev’s later proof in [36] introduced Turaev surfaces and prompted interest in studying their properties.

2. What is the Turaev surface?

Let $D$ be the diagram of a link $L$ drawn on $S^{2}$. For any crossing , we obtain the $A$–smoothing as

$\asymp$

and the $B$–smoothing as $\asymp$. The state $s$ of $D$ is a choice of smoothing at every crossing, resulting in a disjoint union of circles on $S^{2}$. Let $|s|$ denote the number of circles in $s$. Let $s_{A}$ be the all–$A$ state, for which every crossing of $D$ has an $A$–smoothing. Similarly, $s_{B}$ is the all–$B$ state. We will construct the Turaev surface from the dual states $s_{A}$ and $s_{B}$.

At every crossing of $D$, we put a saddle surface which bounds the $A$–smoothing on the top and the $B$–smoothing on the bottom as shown in Figure 1. In this way, we get a cobordism between $s_{A}$ and $s_{B}$, with the link projection $\Gamma$ at the level of the saddles. The Turaev surface $F(D)$ is obtained by attaching $|s_{A}|+|s_{B}|$ discs to all boundary circles. See Figure 1, and [15] for an animation of the Turaev surface for the Borromean link.

The Turaev genus of $D$ is the genus of $F(D)$, given by

[TABLE]

The Turaev genus $g_{T}(L)$ of any non-split link $L$ is the minimum of $g_{T}(D)$ among all diagrams $D$ of $L$. By [36, 12], $L$ is alternating if and only if $g_{T}(L)=0$, and if $D$ is an alternating diagram then $F(D)=S^{2}$. In general, for any link diagram $D$, it follows that (see [12]):

1. (1)

$F(D)$ is a Heegaard surface of $S^{3}$; i.e., an unknotted closed orientable surface in $S^{3}$. 2. (2)

$D$ is alternating on $F(D)$, and the faces of $D$ can be checkerboard colored on $F(D)$, with discs for $s_{A}$ and $s_{B}$ colored white and black, respectively. 3. (3)

$F(D)$ has a Morse decomposition, with $D$ and crossing saddles at height zero, and the $|s_{A}|$ and $|s_{B}|$ discs as maxima and minima, respectively.

Conversely, in [4] conditions were given for a Heegaard surface with cellularly embedded alternating diagram on it to be a Turaev surface.

3. The Turaev surface and the Jones polynomial

A diagram $D$ is $A$–adequate if at each crossing, the two arcs of $s_{A}$ from that crossing are in different state circles. In other words, $|s_{A}|>|s|$ for any state $s$ with exactly one $B$–smoothing. Similarly, we define a $B$–adequate diagram by reversing the roles of $A$ and $B$ above. If $D$ is both $A$–adequate and $B$–adequate it is called adequate. If $D$ is neither $A$–adequate nor $B$–adequate it is called inadequate. A link $L$ is adequate if it has an adequate diagram, and is inadequate if all its diagrams are inadequate. Any reduced alternating diagram is adequate, hence every alternating link is adequate.

Adequacy implies that $s_{A}$ and $s_{B}$ contribute the extreme terms $\pm t^{\alpha}$ and $\pm t^{\beta}$ of the Jones polynomial $V_{L}(t)$, which determine the span $V_{L}(t)=|\alpha-\beta|$, which is a link invariant. Let $c(L)$ be the minimal crossing number among all diagrams for $L$. In [36], Turaev proved

[TABLE]

with equality if $L$ is adequate. If $D$ is a prime non-alternating diagram, then $g_{T}(D)>0$. Thus, span $V_{L}(t)=c(L)$ if and only if $L$ is alternating, from which Tait’s Conjecture follows.

Therefore, for any adequate link $L$ with an adequate diagram $D$ (see [1]),

[TABLE]

So for the connect sum $L\#L^{\prime}$ of adequate links, $g_{T}(L\#L^{\prime})=g_{T}(L)+g_{T}(L^{\prime})$.

Turaev genus and knot homology. Khovanov homology and knot Floer homology categorify the Jones polynomial and the Alexander polynomial, respectively. The width of each bigraded knot homology, $w_{KH}(K)$ and $w_{HF}(K)$, is the number of diagonals with non-zero homology. The Turaev genus bounds the width of both knot homologies [9, 27]:

[TABLE]

For adequate knots, $w_{KH}(K)-2=g_{T}(K)$ [1]. These inequalities have been used to obtain families of knots with unbounded Turaev genus (see [7]).

Ribbon graph invariants. Like the Seifert surface, the Turaev surface provides much more information than its genus. An oriented ribbon graph is a graph with an embedding in an oriented surface, such that its faces are discs. Turaev’s construction determines an oriented ribbon graph $G_{A}$ on $F(D)$: We add an edge for every crossing in $s_{A}$, and collapse each state circle of $s_{A}$ to a vertex of $G_{A}$, preserving the cyclic order of edges given by the checkerboard coloring (see [7]).

If $L$ is alternating, then $V_{L}(t)=T_{G}(-t,-1/t)$, where $T_{G}(x,y)$ is the Tutte polynomial [35]. For any $L$, $V_{L}(t)$ is a specialization of the Bollobás–Riordan–Tutte polynomial of $G_{A}$ [12]. These ideas extend to virtual links and non-orientable ribbon graphs [10]. In [13], a unified description is given for all these knot and ribbon graph polynomial invariants.

4. Turaev genus one links

The Seifert genus is directly computable for alternating and positive links, and has been related to many classical invariants. Moreover, knot Floer homology detects the Seifert genus of knots. In contrast, for most non-adequate links, computing the Turaev genus is an open problem.

The Turaev genus of a link can be computed when the upper bounds in the inequalities (1) or those in [26] match the Turaev genus of a particular diagram, which gives a lower bound. So it is useful to know which diagrams realize a given Turaev genus. Link diagrams with Turaev genus one and two were classified in [5, 22].

This classification uses the decomposition of any prime, connected link diagram $D\subset S^{2}$ into alternating tangles. An edge in $D$ is non-alternating when it joins two overpasses or two underpasses. If $D$ is non-alternating, we can isotope the state circles in $s_{A}$ and $s_{B}$ to intersect exactly at the midpoints of all non-alternating edges of $D$. In the figure to the right from [22], $\alpha\in s_{A},\,\beta\in s_{B}$. The arc $\delta$ joining the points in $\alpha\cap\beta$ is called a cutting arc of $D$.

A cutting arc is the intersection of $S^{2}$ with a compressing disc of the Turaev surface $F(D)$, which intersects $D$ at the endpoints of $\delta$. The boundary $\gamma$ of this compressing disc is called a cutting loop. Every cutting arc of $D$ has a corresponding cutting loop on $F(D)$, and surgery of $D$ along a cutting arc corresponds to surgery of $F(D)$ along a compressing disc, as shown in the following figure from [22].

If $D^{\prime}$ is obtained by surgery from $D$, the surgered surface is its Turaev surface $F(D^{\prime})$ with genus $g_{T}(D^{\prime})=g_{T}(D)-1$. So if $g_{T}(D)=1$, then $\gamma$ is a meridian of the torus $F(D)$, and surgery along all cutting arcs of $D$ cuts the diagram into alternating $2$-tangles [22]. Hence, if $g_{T}(D)=1$, then $D$ is a cycle of alternating $2$-tangles:

This also implies that for any alternating diagram $D$ on its Turaev surface $F(D)$, if $g_{T}(D)\geq 1$ there is an essential simple loop $\gamma$ on $F$ which intersects $D$ twice and bounds a disc in a handlebody bounded by $F$. Thus, the link on the surface in Example 1.3.1 of [26] cannot come from Turaev’s construction. However, this condition is not sufficient; for example, the diagram at right satisfies the condition, but cannot be a Turaev surface because any planar diagram $D$ for this link has more than four crossings, which would remain as crossings on $F(D)$.

Hayashi [16] and Ozawa [30] considered more general ways to quantify the complexity of the pair $(F,D)$, which has prompted recent interest in representativity of knots (see, e.g., [3, 6, 18, 24, 31, 32, 33]).

5. Open problems

Below, we consider open problems in two broad categories:

Question 1**.**

How do you determine the Turaev genus of a knot or link?

Does the Turaev genus always equal the dealternating number of a link? This is true in many cases, and no lower bounds are known to distinguish these invariants (see [26]).

The lower bounds (1) vanish for quasi-alternating links. For any $g>1$, does there exist a quasi-alternating link with Turaev genus $g$?

The Turaev genus is additive under connect sum for adequate knots, and invariant under mutation if the diagram is adequate [1]. In general, for any $K$ and $K^{\prime}$, is $g_{T}(K\#K^{\prime})=g_{T}(K)+g_{T}(K^{\prime})$? If $K$ and $K^{\prime}$ are mutant knots, is $g_{T}(K)=g_{T}(K^{\prime})$? The latter question is open even for adequate knots; if $D$ is a non-adequate diagram of an adequate knot $K$, then for a mutant $D^{\prime}$ of $D$, it might be possible that $g_{T}(K)<g_{T}(D)=g_{T}(D^{\prime})=g_{T}(K^{\prime})$.

If $K$ is a positive knot with Seifert genus $g(K)$, then $g_{T}(K)\leq g(K)$. Is this inequality strict; i.e., is $g_{T}(K)<g(K)$ for a positive knot? It is known to be strict for $g(K)=1,2$ [20, 34] and for adequate positive knots [25].

In general, how do you compute the Turaev genus, which is a link invariant, without using link diagrams? Is it determined by some other link invariants?

Question 2**.**

How do you characterize the Turaev surface?

From the construction in Section 2, it is hard to tell whether a given pair $(F,D)$ is a Turaev surface. The existence of a cutting loop implies that the alternating diagram $D$ on $F$ must have minimal complexity; i.e., there exists an essential simple loop on $F$ which intersects $D$ twice. But this condition is not sufficient. What are the sufficient conditions for a given pair $(F,D)$ to be a Turaev surface?

Alternating, almost alternating and toroidally alternating knots have been characterized topologically using a pair of spanning surfaces in the knot complement [14, 17, 19, 23]. Turaev genus one knots are toroidally alternating, and they contain almost-alternating knots, but they have not been characterized topologically as a separate class of knots. What is a topological characterization of Turaev genus one knots, or generally, of knots with any given Turaev genus?

Any non-split, prime, alternating link in $S^{3}$ is hyperbolic, unless it is a closed 2–braid [28]. This result was recently generalized to links in a thickened surface $F\times I$. If the link $L$ in $F\times I$ admits a diagram on $F$ which is alternating, cellularly embedded, and satisfies an appropriate generalization of “prime,” then $(F\times I)-L$ is hyperbolic [2, 8, 18]. Now, for a given Turaev surface $F(D)$, let $L$ be a link in $F(D)\times I$ which projects to the alternating diagram on $F$. It follows that typically the complement $(F(D)\times I)-L$ is hyperbolic, assuming there are no essential annuli. If $g_{T}(D)=1$ then $(F(D)\times I)-L$ has finite hyperbolic volume. If $g_{T}(D)>1$ then there is a well-defined finite volume if the two boundaries are totally geodesic. How do the geometric invariants of $(F(D)\times I)-L$ depend on the original diagram $D$ in $S^{2}$?