Widths of surface knots
Abstract
We study surface knots in 4–space by using generic planar projections. These projections have fold points and cusps as their singularities and the image of the singular point set divides the plane into several regions. The width (or the total width) of a surface knot is a numerical invariant related to the number of points in the inverse image of a point in each of the regions. We determine the widths of certain surface knots and characterize those surface knots with small total widths. Relation to the surface braid index is also studied.
Yasushi \surnameTakeda \urladdr \volumenumber6 \issuenumber \publicationyear2006 \papernumber64 \startpage1831 \endpage1861 \doi \MR \Zbl \subjectprimarymsc200057Q45 \subjectsecondarymsc200057M25 \published1 November 2006 \publishedonline1 November 2006 \proposed \seconded \corresponding \editorCPR \version \arxivreference
1 Introduction
The notion of width for classical knots was introduced by Gabai [8] as a generalization of the bridge index, which plays an important role in the classical knot theory. The width was useful for solving difficult problems. More precisely, we consider a generic projection of an embedded circle in into the line as in \fullreffig:a. Then nondegenerate critical points appear as its singularities and their images divide the line into several intervals. For each such interval, we consider the number of points in for a point in the interval, and we call it the local width, which does not depend on the choice of . The width of a knot is the minimum of the total of local widths over all embedded circles representing the given knot.
By a surface knot, we mean (the isotopy class of) a closed connected (possibly nonorientable) smoothly embedded surface in . For a surface knot, Carter–Saito [5, Section 4.6] considered the analogy of the width. They applied the notion of chart for the definition of width for surface knots. A chart is a planar projection of a surface knot together with an associated graph, which was first introduced in the surface braid theory (see Kamada [12]). The graph is constructed by using a generic projection into –space of a surface knot. The generic projections into –space of surface knots have double points, triple points, and branch points as their singularities, and the charts represent the state of the combination of these singularities. Moreover, charts form several planar regions which are surrounded by curves representing double points and fold lines, and the width of a surface knot which Carter–Saito introduced is defined by using the number of points in the fiber over a point (local width) in each of these regions like the width for classical knots. They considered the minimum (over all representatives of the given isotopy class) of the maximum of local widths over all the regions.
However, the width which Carter–Saito defined is slightly different from the one which Gabai defined. In fact, Carter–Saito considered the maximum of local widths for the definition of width and Gabai considered the total of local widths. Moreover, the width of surface knots has not been studied so much until now as far as the author knows.
In this paper, for surface knots, we study the width defined by Carter–Saito, and the total width which is the straightforward analogy of the width for classical knots defined by Gabai. For this purpose, we consider generic planar projections of surface knots instead of charts. In the surface knot theory, we often use generic projections into –space: in fact, many results have been obtainted by using projections into –space, and since we can view the diagrams in –space, they facilitate the study of surface knots. Generic planar projections have also been useful (for example, see Carrara, Carter and Saito [3], Carrara, Ruas, and Saeki [4], Saeki and Takeda [16] and Yamamoto [22]). Planar projections have fold points and cusps as their singularities. Cusps appear as discrete points and fold points appear as a –dimensional submanifold. Let us call the set of cusps and fold points in the surface the singular set. For a given surface knot, the image of the singular set divides the plane into several regions. For each such region, we consider the number of points in the preimage of a point in that region and the maximum or the total of these numbers over all the regions. Then we take the minimum of these numbers over all embedded surfaces representing the given surface knot. Roughly speaking, this defines the width and the total width of a surface knot.
The paper is organized as follows. In \fullrefsec2 we define the width and the total width of surface knots and recall the definitions of the genericity of mappings and the triviality of surface knots. In \fullrefsec3 we study the width and determine the width of some surface knots such as ribbon surface knots and –twist spun –bridge knots. In \fullrefsec4 we consider the relationship between the width and the surface braid index and show that the width is always smaller than or equal to the twice of the surface braid index plus two. We also show that in general the difference between these two invariants can be arbitrarily large. In \fullrefsec5 we give some characterization theorems of surface knots with small total widths.
Throughout the paper, we work in the smooth category.
The author would like to thank Professor Osamu Saeki for helpful suggestions and Professor Mitsuyoshi Kato for his constant encouragement. He also thanks the referee for careful reading and useful comments. The author has been supported by JSPS Research Fellowships for Young Scientists.
2 Preliminaries
In this section, we prepare several notions from singularity theory and define the width and the total width for surface knots in . For singularity theory, the reader is referred to Golubitsky and Guillemin [9], for example.
Definition 2.1.
Let be a closed connected surface. Denote by the set of all smooth mappings from to , endowed with the Whitney topology. Let and be elements of . Then is equivalent to if there exist diffeomorphisms and such that .
Definition 2.2.
Let be an element of . Then is said to be stable if there exists a neighborhood of in such that each in is equivalent to .
Definition 2.3.
Let be a smooth mapping from to . Then is called a fold point if we can choose local coordinates centered at and centered at such that , in a neighborhood of , is of the form:
Moreover, is called a cusp if we can choose local coordinates centered at and centered at such that , in a neighborhood of , is of the form:
We denote by the set of fold points and cusps, and by the set of cusps.
Note that is a regular –dimensional submanifold of and is a discrete set.
Recall the following wellknown characterization of stable mappings in.
Proposition 2.1.
Let be a smooth mapping from a closed connected surface to . Then is stable if and only if has only fold points and cusps as its singularities, its restriction to the set of fold points is an immersion with normal crossings, and for each cusp , we have:
Let be a closed connected surface. For a smooth map , we set
which is called the singular point set of . If is stable, then we clearly have .
The following theorem is wellknown (see Thom [20]).
Theorem 2.1.
Let be a stable mapping from a closed connected surface to . Then the number of cusps of has the same parity as the Euler characteristic of .
Definition 2.4.
Let be an embedding of a closed connected surface. Let be an orthogonal projection. Then we say that is generic with respect to (or with respect to ) if is stable.
By Mather [15], almost every orthogonal projection is generic with respect to .
Definition 2.5.
Let be an embedding of a closed connected surface . Let be an orthogonal projection which is generic with respect to . In this cace, has fold points and cusps as its singularities. Let () denote the set of these singularities. The singular value set divides the plane into several regions. For a point in a given region, we call the number of elements in the set the local width, which does not depend on the choice of and is always even (see the proof of \fullreflem:t1). Let (or ) be the maximum of the local widths over all the regions and (or ) be the total of the local widths over all the regions. The width of a surface knot is the minimum of , where runs over all embeddings isotopic to and runs over all orthogonal projections which are generic with respect to . Moreover, the total width of a surface knot is the minimum of , where runs over all embeddings isotopic to and runs over all orthogonal projections which are generic with respect to .
Let us now recall the definitions of a handlebody, the standard projective planes in and the normal Euler number.
An orientable handlebody is a compact orientable –manifold obtained by attaching a finite number of –handles to a –ball (the number of –handles may possibly be zero). A nonorientable handlebody is a compact nonorientable –manifold obtained by attaching a finite number of –handles to a –ball.
The standardly embedded projective plane in is constructed as in \fullreffig:b, by attaching an unknotted disk in to a “trivially embedded” band in . We have two trivially embedded bands up to isotopy, and accordingly we have two kinds of standard projective planes in . These surface knots have normal Euler number . Normal Euler number is an isotopy invariant of surface knots (for example, see Carter and Saito [5]).
There are several definitions of trivial surface knots in the litterature (for example, see Hosokawa and Kawauchi [10]). In this paper, we adopt the following definition.
Definition 2.6.
For a surface knot, we say that it is strongly trivial if it is the boundary of a handlebody embedded in . Moreover, we say that a nonorientable surface knot is trivial if it is the connected sum of some copies of the standardly embedded projective planes in , that is, the connected sum of copies of the standardly embedded projective plane with normal Euler number and copies with normal Euler number for some and with .
A surface knot is trivial if it is strongly trivial. However, a trivial surface knot may not necessarily be strongly trivial. In fact, if a surface knot is strongly trivial, then its Euler characteristic must be even. More precisely, a trivial surface knot is strongly trivial if and only if its normal Euler number vanishes. Furthermore, for a closed connected nonorientable surface of nonorientable genus , the number of trivial surface knots diffeomorphic to it is equal to , and if is even, then a strongly trivial surface knot diffeomorphic to it exists and is unique (for example, see [10]).
The following lemma is often used throughout this paper.
Lemma 2.1 (Carrara, Ruas and Saeki [4]).
Let and be orthogonal projections. For an embedding of a closed connected surface , if is stable without cusps and is a Morse function^{1}^{1}1A smooth function on a smooth manifold is a Morse function if its critical points are all nondegenerate. with at most four critical points, then is strongly trivial.
3 Widths of certain surface knots
In this section, we characterize those surface knots with width two and determine the widths of ribbon surface knots and –twist spun –bridge knots.
Let we begin by the following lemma.
Lemma 3.1.
Let be a closed connected surface and be an embedding. Let be an orthogonal projection which is generic with respect to . Suppose that there exists a proper arc in isotopic to a line in such that intersects transversely at two points both of which are the images of fold points. Let be a tubular neighborhood of in and let and be the connected components of Int. Then there exist embeddings of closed connected surfaces into , , such that

is isotopic to the connected sum ,

is generic with respect to , ,

,

for , there exists a –disk such that

,

,


for , is a mapping as depicted in \fullreffig:c.
Proof.
Set . Then is a closed –dimensional manifold, and the embedding into is a trivial knot, since is a Morse function with one maximum and one minimum. Therefore, bounds a 2–disk in , . We slightly push the interior of the –disk into and we denote it by . Then we get the desired embeddings , such that and . ∎
Let be an embedding of a closed connected surface and be an orthogonal projection which is generic with respect to . Then has fold crossings and cusps. We have four regions locally near a fold crossing, and we have two regions locally near a cusp.
Lemma 3.2.
Let be an embedding of a closed connected surface and be an orthogonal projection which is generic with respect to . Then, the local widths around a fold crossing of are of the forms for some even. The local widths around the image of a cusp are of the forms for some even. See \fullreffig:d.
Proof.
If a point crosses the image of a fold curve, then the number of elements in the inverse image changes by . Furthermore, since is compact, is not surjective, and the local width for the unbounded region must be zero. Therefore, the local width of each region should be an even number.
Let be a fold crossing. Then the mapping near is easily seen to be equivalent to the mapping as depicted in \fullreffig:e for some . Furthermore, each local width should be even. Therefore, the desired conclusion follows.
For a cusp, the situation is as depicted in \fullreffig:f for some . Since the mapping near a cusp point is an open map, each local width around the image of a cusp should be positive. Then, the desired conclusion follows. This completes the proof. ∎
Let us give a characterization of surface knots with width two.
Theorem 3.1.
Let be a surface knot. Then if and only if is strongly trivial.
Proof.
Suppose that . We may assume that for an orthogonal projection which is generic with respect to , we have , where is the inclusion mapping. Then the local width of each region of must be equal to or . Therefore, by \fullreflem:t1 there are no fold crossings nor cusps. Since is connected, we see that the image of the singular set must be as depicted in \fullreffig:g up to isotopy of . Then by using \fullreflem:t0, we see that either (i) is isotopic to a connected sum for some such that is generic with respect to and the image of the singular set is as depicted in \fullreffig:h up to isotopy of , , or (ii) the image of the singular set is as depicted in \fullreffig:i up to isotopy of . In case (i), each is strongly trivial by \fullreflem:crs. Therefore, is also strongly trivial. In case (ii), is strongly trivial by \fullreflem:crs. Conversely, if is strongly trivial, then we see easily that . This completes the proof. ∎
Let us recall the notion of a ribbon surface knot, which plays an important role in the theory of surface knots (Cochran, Kamada, Kawauchi, Tanaka and Yasuda [6, 11, 13, 18, 23]).
Definition 3.1.
Let (or ) denote a finite disjoint collection of –balls embedded in . Parametrize each component of as . Suppose that for each , we have

, and

for a finite set .
Then the surface knot
(after a suitable smoothing) is called a ribbon surface knot if is connected.
Note that a surface knot which is strongly trivial is a ribbon surface knot. If a ribbon surface knot is nonorientable, then the genus must be even.
Proposition 3.1.
Let be a ribbon surface knot which is not strongly trivial. Then we have .
Proof.
By isotopy of we may assume that
We define by . Then is generic for and is as depicted in \fullreffig:j. Moreover, we may further assume that each satisfies and is an embedding into the closure of , where and is sufficiently small.
We define
by , where is a copy of , .
We may assume that restricted to is an immersion with normal crossings. Furthermore, by pushing the crossings out of one by one by an isotopy of , we may assume that does not contain any double point of restricted to (see \fullreffig:k).
Now the fiber of the normal disk bundle to in is a –dimensional disk. If we fix , then the isotopy class of is determined by the homotopy class of a unit normal vector field along , which corresponds to the unit normal vector to in the –dimensional disk fiber. Therefore, we may assume that the tangent plane to at is not parallel to the fibers of , .
By taking “thin” enough, we may then assume that consists exactly of two arcs for each . Now is as depicted in \fullreffig:l and we see that the local width of each region is equal to or . Therefore, we have . Then by \fullrefthm:t0, the desired conclusion follows. This completes the proof. ∎
Let us recall the notion of bridge index for classical knots. Here, we give a definition suitable for our purpose.
Definition 3.2.
Let be a classical knot and a generic orthogonal projection. Let be the number of local maxima of . Then the bridge index of is defined to be the minimum of , where runs through all embeddings of into isotopic to , and runs through all orthogonal projections generic with respect to . A knot having bridge index is called an –bridge knot.
Note that an orthogonal projection is generic with respect to if has only nondegenerate critical points as its singularities.
Definition 3.3.
Let be the –dimensional upper halfspace, ie,
and the plane . Let be an arc properly embedded in the halfspace . When the halfspace is rotated around the plane in , the continuous trace of forms a –sphere. This –sphere is said to be derived from by (untwisted) spinning, and we call the resulting surface knot a spun knot. Moreover, put the knotted part of in a –ball as in \fullreffig:m and twist it times, , as the halfspace spins once around . Then we call the resulting surface knot an –twist spun knot. In general, is associated with a knot in , which is obtained by connecting the end points of in an obvious way by an arc in . See also Zeeman [24].
Proposition 3.2.
Let be an –twist spun –bridge knot with . Then we have .
Proof.
Let be a –bridge knot and the orthogonal projection defined by . Then there exists a knot isotopic to such that is generic for and has two local minima and two local maxima with . We may assume that the values of the local maxima and the local minima are all distinct and that is in a position as described in \fullreffig:n. Rotate the part around in . Then we get the –twist spun of . The orthogonal projection defined by is generic for and is as depicted in \fullreffig:o. Therefore, we have . For the –twist spun of , rotate around once and twist the “knotted part” times. Then does not change and the image of the singular set is again as depicted in \fullreffig:o. Therefore, we have . If , then by \fullrefthm:t0 is strongly trivial. However, for , it is known that is not strongly trivial (Cochran [6]). Therefore, we have for . ∎
Remark 3.1.
By an argument similar to that in the proof of \fullrefprop:t1, we can show that the width of an –twist spun –bridge knot is smaller than or equal to . However, even if , the equality may not hold. In fact, for every knot, its –twist spun is a ribbon surface knot (see, for example, [6]). Hence, by \fullrefprop:t0, we have if is a –twist spun –bridge knot with .
4 Braid index and width
In this section, we study the relationship between the braid index and the width of a surface knot. Throughout this section, we assume that surface knots are orientable.
The notion of surface braid was introduced by Kamada [12]. Kamada and Viro showed that every orientable surface knot is isotopic to a simple closed surface braid.
A closed surface braid in is a closed oriented surface embedded in such that the restriction map of the projection to the second factor is an orientation preserving branched covering. We say that it is a simple closed surface braid if is a simple branched covering. An orientation preserving branched covering between closed oriented surfaces is simple if for every branch point , we have , where denotes the number of elements and deg() is the mapping degree of . The mapping degree of is called the degree of the closed surface braid.
The braid index Braid() of an oriented surface knot in is the minimum degree of simple closed surface braids in that are isotopic to .
For classical knots, the bridge index is smaller than or equal to the braid index. On the other hand, the relation between the width and the braid index for classical knots has not been studied as far as the author knows.
For surface knots, we have the following.
Proposition 4.1.
Let be an orientable surface knot. Then we have
Proof.
Let be the standard –sphere, ie, , and be its tubular neighborhood. We may assume that and the restriction of is a simple branched covering of degree equal to Braid(). We may further assume that the critical values of all lie near and that outside of the preimage of a neighborbhood of , is almost parallel to . Let us define the orthogonal projection by . Then, we may assume that the image of the singular points of is as depicted in \fullreffig:p.
Let be a branch point of and let be the branch point such that . Furthermore, let be a small neighborhood of in , where and corresponds to (0,0), and let be the component of which contains . Set for . Then can be regarded as a –string braid for . See \fullreffig:q (1).
Then we deform (or more precisely, we deform ) by an isotopy in so that this sequence of –string braids is deformed as in \fullreffig:q (2). Note that then is generic on and the image of the singular points in is as depicted in \fullreffig:q (3). Three cusps are created, while the branch point in question is eliminated.
We perform the above described deformation for each branch point of . Then we get a surface isotopic to such that is generic with respect to and that the singular values of and the local widths are as depicted in \fullreffig:r, where Braid(). Therefore, we have . This completes the proof. ∎
Let us consider a branch point of a surface braid as above. Since it is simple, there may be a “sheet” of over that point which does not intersect a neighborhood of the corresponding branch point in . If the sheet can be deformed as depicted in \fullreffig:r2, then the width decreases by . Therefore, the following conjecture seems to be plausible.
Conjecture 4.1.
Let be an orientable surface knot. Then we have
By the following proposition, the difference between the width and (twice) the braid index can be arbitrarily large.
Proposition 4.2.
For every , there exists a surface knot in with Braid() and .
For the proof, we need the following.
Lemma 4.1.
For surface knots and in , we always have
Proof.
We may assume that there exists an orthogonal projection which is generic with respect to both and such that and . We may further assume that . Let us consider fold points of and whose images by lie in the outermost boundaries of and respectively. If we perform the connected sum operation using small disk neighborhoods of these fold points and by connecting and by an appropriate cylinder (see the proof of \fullrefprop:t0), then is generic with respect to and . Thus the conclusion follows. This completes the proof. ∎
Remark 4.1.
The referee kindly pointed out that there is an example for which the equality does not hold in \fullreflem:t2 as follows. By Viro [21], it is known that there exists a ribbon –sphere knot , which is not strongly trivial, such that is isotopic to , where is the trivial projective plane with normal Euler number . Let be a Klein bottle knot, which is strongly trivial, such that is isotopic to , where is the trivial projective plane with normal Euler number . Then is isotopic to . Since , and , the equality does not hold in \fullreflem:t2. However, we do not know such an example if both and are orientable. The author would like to thank the referee for pointing out this example.
Proof of \fullrefprop:t9.
Let be the spun ()–torus knot, where is an odd integer with . Furthermore, let be the connected sum of copies of . Then by Tanaka [19], we have Braid() . On the other hand, since the ()–torus knot is a –bridge knot, we have by \fullrefprop:t1. Then by \fullreflem:t2, we have . Since Braid() , is not strongly trivial, and hence by \fullrefthm:t0. Therefore, we have . This completes the proof. ∎
5 Total widths of surface knots
In this section, we give several characterization theorems of surface knots with small total widths.
The following is an immediate consequence of \fullrefthm:t0.
Theorem 5.1.
Let be a surface knot. Then if and only if it is strongly trivial.
Let be a stable mapping of a closed surface into the plane. For a point , we give a local orientation of at as follows. For a sufficiently small disk neighborhood of in , is an arc and consists of two regions. Let us take points, say and , from each of the two regions. We may assume that the number of elements in the inverse image is greater than that of . Then we orient so that the left hand side region corresponds to . Finally we give a local orientation of at so that preserves the orientation around . See \fullreffig:s.
It is easy to see that the above local orientations vary continuously and that they define a globally welldefined orientation on .
On the other hand, by considering the “line” for each , we obtain a smooth mapping P. It is not difficult to see that this mapping extends to a smooth mapping P. We orient P so that the lines rotating in the counterclockwise direction correspond to the positive direction of P.
Then we define rot() to be the mapping degree of P.
Then the following lemma is proved in Levine [14].
Lemma 5.1.
The Euler characteristic of coincides with rot().
Using \fullreflem:le, we prove the following.
Theorem 5.2.
Let be a surface knot which is diffeomorphic to the –sphere . Then if and only if it is strongly trivial.
Proof.
If , then by \fullrefthm:t1, is strongly trivial. Furthermore, there does not exist a surface knot with , since is connected. Therefore, we may assume and there exists an orthogonal projection which is generic with respect to such that .
If has no fold crossings, then it is of the form “Type ” as depicted in \fullreffig:t up to isotopy of . Then, by \fullreflem:t0, is the connected sum of surface knots and such that (or ) is of the form “Type ” (resp. “Type ”) as depicted in \fullreffig:t up to isotopy of . Since is diffeomorphic to the –sphere, so are and . Then, by \fullreflem:le, Type and Type must correspond to Type and Type of \fullreffig:t respectively. By \fullreflem:crs, we see that is strongly trivial. Furthermore, there exists an orthogonal projection which is generic with respect to such that has exactly two critical points. In fact, such a projection can be obtained by composing and a suitable projection (for example, see Fukuda [7]). Thus, is also strongly trivial, and hence so is .
If has one fold crossing, then it is of the form “Type ” as depicted in \fullreffig:u or in \fullreffig:v up to isotopy of . Then, by \fullreflem:t0, is the connected sum of surface knots and such that (or ) is of the form “Type ” (resp. “Type ”) as depicted in \fullreffig:u or in \fullreffig:v up to isotopy of . Since is diffeomorphic to the –sphere, so are and . Then, by \fullreflem:le, Type and Type must correspond to Type and Type of \fullreffig:u or \fullreffig:v respectively. By \fullreflem:crs, is strongly trivial. Furthermore, there exists an orthogonal projection which is generic with respect to such that has exactly four critical points. Therefore, is strongly trivial by Scharlemann [17]. (In fact, “Type ” of \fullreffig:v does not occur by Akhmet’ev [1, 23. Corollary].) Thus, is strongly trivial.
If has two fold crossings, then it is of the form “Type ” as depicted in \fullreffig:w or as depicted in \fullreffig:x. In the former case, we see that is strongly trivial as before (see \fullreffig:w). In the latter case, we see that by \fullreflem:le, which is a contradiction. Thus, this case does not occur.
If has three or more fold crossings, then it is of the form as depicted in \fullreffig:y. Then, we see that by \fullreflem:le, so that this case does not occur.
Hence is always strongly trivial. This completes the proof. ∎
Corollary 5.1.
Let be an –twist spun 2–bridge knot with . Then we have .
Proof.
Since is not strongly trivial, by \fullrefthm:t2 we have . On the other hand, since has planar projection as in \fullreffig:o, we have . This completes the proof. ∎
Similarly, for surface knots diffeomorphic to the projective plane, we have the following characterization.
Theorem 5.3.
Let be a surface knot which is diffeomorphic to the projective plane P. Then if and only if it is trivial.
Proof.
If , then by \fullrefthm:t1, is strongly trivial. Furthermore, there does not exist a surface knot with , since is connected. Therefore, we may assume and there exists an orthogonal projection which is generic with respect to such that .
We use the argument of the proof of \fullrefthm:t2. If has no fold crossings, then it is of the form “Type ” as depicted in \fullreffig:t. Since is diffeomorphic to the projective plane, by \fullreflem:t0 we see that , where is of the form “Type ” as depicted in \fullreffig:t and is of the form “Type ” as depicted in \fullreffig:z. By \fullreflem:crs, is strongly trivial. Since there exists an orthogonal projection which is generic with respect to such that has exactly three critical points, we see that