A characterization of almost alternating knots
Generalizing Howie and Greene’s characterization of alternating knots, we give a topological characterization of almost alternating knots.
Key words and phrases:Almost alternating knot, spanning surface defect, alternating genus
2010 Mathematics Subject Classification:Primary 57M25 , Secondary 57M27
Recently, an intrinsic characterization of alternating knots and links in was given by Howie [Ho1, Ho2] and Greene [Gr] by using a spanning surface of a knot , an embedded connected compact surface with that is not necessarily orientable.
A knot in is alternating if and only if has spanning surfaces and satisfying one of the following properties.
For a spanning surface , the Gordon-Litherland pairing is the symmetric bilinear form on defined by . Here is the double covering from the unit normal of , and and denote (multi) curves on that represent and . Let be the euler number of , which is equal to the twice of the boundary slope of . Since [GL, Corollary 5], where denotes the signature of , two characterizations are essentially the same although they look different at first glance.
Inspired from Howie and Greene’s argument, we explore a similar characterization for almost alternating knots. A knot is almost alternating if is represented by an almost alternating diagram, a diagram such that a single crossing change makes the diagram alternating (see [A] for its basic properties). In this paper we regard an alternating knot as a special case of an almost alternating knot.
For a spanning surface , we say that an embedded disk in is an almost compressing disk of , if has the following properties.
The knot transversely intersect with the interior of at one point.
transversely intersects with , except one point . At the point , has a saddle tangency with .
The intersection is a union of and a simple arc connecting and the unique intersection point . We call this arc the intersection arc of (See Figure 1 (a)).
A typical situation where an almost compressing disk appears is a checkerboard surface of a knot diagram with a reducible crossing. A crossing in a knot diagram is a reducible crossing if there is a circle in the projection plane which transversely intersects the diagram at one point . Let be the disk bounded by such a circle lying in the upper half space. After a slight perturbation near , gives an almost compressing disk of a checkerboard surface of the diagram (See Figure 1 (b)).
Using spanning surfaces, their Gordon-Litherland pairings and almost compressing disks, our characterization of almost alternating knots is stated as follows. (Although our characterization can be generalized for links (See Remark 6), throughout the paper we will mainly treat a knot case for sake of simplicity.)
A knot in is almost alternating if and only if has spanning surfaces and which intersect transversely, such that
There exist an almost compressing disk of and an almost compressing disk of such that
transversely intersects at exactly one clasp intersection.
The clasp intersection is equal to .
The union of intersection arc is contained in .
One can understand the condition (ii) as follows. See Figure 2 (a) for an illustration of almost compressing disks satisfying the condition (ii-a) and (ii-b). Near the clasp intersection , two spanning surfaces and appear so that the condition (ii-c) is satisfied. Figure 2 (b) gives a local model for surfaces and that satisfy the condition (ii).
The spanning surfaces and are mutually intersecting twisted bands. Their almost compressing disks and appear as half planes, slightly perturbed so that they form a clasp intersection.
To understand Figure 2 (b), we take a local coordinate and consider a sequence of slices by the horizontal planes, as shown in Figure 2 (c). As the height increases, two points which are the the slice of the knot turn. The slice of spanning surfaces and turn accordingly. At the critical moment , the intersection of and appears as a vertical line segment. In the sequence of slice, appears as a family of vertical half lines. At , overlaps with and the intersection arc appears. Also, appears as a half-plane in the critical level . By chasing the movie of slices, we see that at , and forms a clasp intersection which is equal to . Moreover, in the slice , the union of intersection arcs coincides with the vertical line segment .
These almost compressing disks and spanning surfaces appear from an alternating diagram on the torus coming from an almost alternating diagram. For an almost alternating diagram we add a one-handle near the almost alternating crossing . By pushing the almost alternating crossing to the one-handle, we reverse the over-under information at to get an alternating diagram on the standardly embedded torus (see Figure 3 (a,b)).
Let and be the checkerboard surface from the resulting alternating diagram on the torus . Take a meridian a (the boundary of a co-core of ) and longitude of the torus so that they intersect exactly once at the crossing point . The disks bounded by these curves give rise to an almost compressing disk and . Near the common point we slightly push and so that they form a clasp intersection (see Figure 3 (c)). Then the surfaces and near are the same as our local model in Figure 2 (b) so they satisfy the condition (ii) in Theorem 2.
As these arguments demonstrate, our proof of Theorem 2 comes from a point of view that an almost alternating knot is a special case of a toroidally alternating knot, a knot admitting a cellular alternating projection on a standardly embedded torus [Ad]. Here we say that a knot diagram on a surface is cellular if it cuts into a disjoint union of disks.
By Theorem 1, in a setting of Theorem 2 if either or is compressible then is alternating. The almost compressing disk condition (ii) says that both and are ‘close’ to compressible and encodes where is an almost alternating crossing in terms of the almost compressing disks.
Although in this point of view, it is natural to expect the condition (i) in Theorem 2 is equivalent to toroidally alternating, it is not the case. (See Remark 5). Nevertheless, the quantity is interesting in its own right. For a knot we define the spanning surface defect by
Let be the alternating genus of , the minimum genus of a Heegaard surface in such that has a cellular knot diagram on . We will show the following inequality.
The author was partially supported by JSPS KAKENHI, Grant Number 15K17540. He gratefully thanks to Joshua Howie and Joshua Greene for pointing out an error in the first version. He sincerely wish to thank Joshua Howie for sharing his deep insight. He also thanks to Tetsuya Abe for comments on earlier draft of the paper.
Let be a knot which is represented by a diagram on an oriented embedded closed surface . Assume that the diagram admits a checkerboard coloring. That is, for each complementary region of the diagram one can associate the black or white colors so that no two adjacent regions have the same color. By attaching twisted bands at the corners of black-colored (resp. white-colored) regions, we get a spanning surface (resp. ) of which we call the checkerboard surfaces.
We say that a crossing of a diagram is of type a (resp. type b) if near the crossing , the coloring is as shown in Figure 4 (1). Also, we orient the diagram and we say that is of type I (resp. type II) if near the crossing , coloring and orientations are as shown in Figure 4 (2).
Let be the number of the crossings of the diagram . We denote the number of the crossings of type and , by and . respectively.
Let and be the checkerboard surface of a knot coming from a cellular knot diagram on a surface .
At least one of or is non-orientable.
(i): Fix a checkerboard coloring of . Then the notion of type I and type II crossings does not depend on a choice of an orientation of . If the diagram has a crossing of type I (resp. type II), then (resp. ) is non-orientable.
(ii): A cellular diagram induces a cellular decomposition of whose -, - and -cells correspond to the black colored regions, the crossings, and the white colored regions, respectively. By definition, and . Therefore
(iii): Let be the blackboard framing of , the framing determined by . By comparing the framings determined by the checkerboard surfaces and the blackboard framing (see Figure 5), we get
Here denotes the number of crossings which is both of type a and of type I. The meanings of and are similar. Therefore
Lemma 1 gives an estimate of the crosscap number of a knot , the minimum 1st betti number of a non-orientable spanning surface of .
If a knot is represented by a cellular diagram on a surface which admits a checkerboard coloring, then . Here denotes the ceiling of , the minimum integer which is greater than or equal to .
In a case (a usual knot diagram), a slightly better bound is known [MY]. Here denotes the floor of , the maximum integer which is smaller than or equal to . Indeed, it is easy to see that a slightly better inequality holds unless .
A cellular diagram may not admit a checkerboard coloring in general. The following is an interesting property of an alternating cellular diagram.
If a link diagram on a surface is cellular and alternating, then it admits a checkerboard coloring.
Take simple closed curves cutting the surface into a -gon so that they are disjoint from the crossings of and transverse to . At each , we assign the symbol (over) or (under) according to the over-under information of the arc in that passes . The alternating assumption of implies that the symbol and alternate along . The cellular assumption of implies that for all .
Assume to the contrary that admits no checkerboard coloring. If all of are even, a checkerboard coloring of extends to a checkerboard coloring of . Thus we may assume that is odd. Since , when we glue along to recover , the points with the same symbol is identified (see Figure 6). This contradicts the assumption that is alternating.
Now we are ready to prove our main theorems.
Proof of Theorem 3.
The proof of Theorem 3 does not use the property that is a Heegaard surface. We actually show that gives a lower bound of the minimum genus of closed embedded surface which is not necessarily a Heegaard surface such that admits a cellular alternating diagram on .
Proof of Theorem 2.
First we prove the ‘only if’ part. The property (i) follows from Theorem 3. As we have already mentioned in Introduction, an almost alternating diagram of yields a cellular alternating diagram on the standardly embedded torus, and we have the checkerboard surfaces and , their almost compressing disks and having the property (ii).
We prove the ‘if’ part. By Theorem 1, if then is alternating. Hence throughout the proof, we will assume that
First we note that this assumption leads to the following.
Both and are incompressible.
Proof of Claim 1.
If or is compressible then compression gives new spanning surfaces and with . This contradicts with (D). ∎
We fix a tubular neighborhood of , and let . We put and . Similarly, we put and . In condition (ii) we are assuming that and intersect transversely. With no loss of generality, we will always assume the following additional transversality properties.
and are simple closed curves on that intersect efficiently.
intersects with the almost compressing disk of transversely. Similarly, intersects with the almost compressing disk of transversely.
Here we say that two curves intersects efficiently if they are transverse and attain the minimal geometric intersection.
Our next task is to simplify the intersections of and .
We can put and so that and hold.
Proof of Claim 2.
By our assumption (ii) and transversality (T2), on the almost compressing disk , the connected components of are classified into the following three types.
A simple closed curve in .
An embedded arc connecting two points on .
An arc connecting a point and a point on .
By our assumption (ii-c), , so . This shows that a component of type (c) is nothing but the intersection arc .
A simple closed curve component of type (a) bounds a disk in . Take an innermost so that the interior of is disjoint from . Since is incompressible by Claim 1, bounds a disk in and is a 2-sphere bounding a 3-ball in .
We show that . Assume to the contrary that . Then so it bounds a subdisk . Then gives a sphere that transversely intersect with the knot at exactly one point, which is impossible.
Therefore we can remove by surgerying along (see Figure 7 (1)), that is, by pushing across the 3-ball . Since , this surgery does not affect so the condition (ii) is still satisfied. This shows that one can remove all the type (a) components of .
Similarly, an arc component of type (b) cuts into a smaller disk that does not contain . Take an outermost so that the interior of is disjoint from . Then we push along to remove the intersection (see Figure 7 (2)). Since , this surgery does not affect so the condition (ii) is still satisfied.
Therefore we conclude that one can remove all the type (a) and (b) components of . Thus consisits of a connected component of type (c) so .
By the same argument we put so that .
Since and are embedded, by transversality each connected component of is either an arc or a simple closed curve. We denote by (resp. ) the set of the connected components of which is an arc (resp. a simple closed curve).
One can put and so that no circle bounds a disk in or , preserving the property and .
Assume to the contrary that a circle bounds a disk in (the case bounds a disk in is similar). Take an innermost one so that the interior of is disjoint from . Since is incompressible by Claim 1, bounds a disk and is a 2-sphere bounding a 3-ball in . By the same argument as Claim 2, is disjoint from .
By Claim 2, . This shows that is contained in an arc in . Since for any arc , this shows that . Therefore the 3-ball is also disjoint from .
Thus by surgerying along (by pushing across the 3-ball , we can remove the intersection circle , without affecting and . ∎
Each arc component of naturally extends to an arc in connecting two different points in the knot which we denote by . By [Ho1, Lemma 1], such an intersection arc is ‘standard’, in the following sense.
Each arc is locally homeomorphic to the intersections of two checkerboard surface near the famous “Menasco crossing ball” (see Figure 8.)
Thus by collapsing arcs to a point, we get an immersed surface .
Proof of 4.
By the transversality assumption (T1) and the definition of the euler number, . Therefore
Hence we conclude
By our assumption (i) and (D), .
On the other hand, for a spanning surface of a knot , the Gordon-Litherland pairing is non-degenerate so (mod ). Therefore must be even hence . ∎
We denote the collapsing map by and its extension to ambient space by . By abuse of notation, we denote the image of double point circles by the same symbol .
Let and be the four-valent graph in . We assign the black or white color on each complementary region so that no two adjacent regions have the same color, and that and holds. Here denotes the black and white colored regions of .
By assumption (ii-c), so it is identified with for some . Thus is sent to a four-valent vertex of under the collapsing map . We consider the simple closed curves , . By Claim 2, they have the following properties.
and transversely intersect at exactly one point , and
and are disjoint from double point circles .
is an embedded torus.
Proof of Claim 5.
By Claim 4, is a torus or a Klein bottle. Assume to the contrary that is not embedded. By Claim 3 every preimage of a double point circle is an essential simple closed curve in . Hence they cut into annuli or Möbius bands . The graph is connected and disjoint from double point circles. Hence it is contained in exactly one component, say . Thus we may assume that all the other components are contained in a black colored region . The double point curve is an intersection of and so the number of preimage of double point circles in and in must be the same. This forces to and is an annulus.
On the other hand, by (B1) and (B2), are simple closed curves in that transversely intersect at exactly one point . This is a contradiction because an annulus cannot contain such simple closed curves. Since the Klein bottle cannot be embedded into , is an embedded torus. ∎
By construction of the surface , the knot is contained in a neighborhood of . By (T1), the projection of on yields an alternating diagram on . By (B1) and (B2), and are simple closed curves which cuts into a disk. Moreover by Claim 2 they bound disks and in whose interiors are disjoint from . Thus is a standard embedding. Also, the diagram on is cellular, because otherwise there exists a simple closed curve contained in or which is isotopic to or . Existence of such loop implies or is compressible, which contradicts with Claim 1.
Now by cutting along and we get an alternating tangle in the square, and the diagram on can be recovered by gluing their edges (see Figure 9 (a,b)). The resulting alternating diagram on a torus can be seen as an almost alternating diagram Figure 9 (c), with almost alternating crossing which corresponds to .
As one can see in the proof, we used almost compressing disks not only to control the alternating diagram on torus , but also to prove Claim 5, the surface constructed from spanning surfaces and is an embedded torus. In fact, Howie presents a construction of knot with spanning surfaces and such that and that they do not give rise to an alternating diagram on embedded torus [Ho2].
Toroidally alternating links and their generalizations, a knot admitting alternating diagram on surface with several additional properties, are extensively discussed in [Ho2].
In our proof, we used the assumption that is a knot, only to guarantee the property (St) of the standardness of the intersections of and . As noted in [Ho1, Ho2], for a non-split link case, to guarantee the standardness (St) it is sufficient to add an assumption that
The intersection of and have the same sign.
Thus, by adding the assumption (iii) we have a characterization of almost alternating links as well.
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