PPOSITIVE ORIENTED THOMPSON LINKS
VALERIANO AIELLO AND SEBASTIAN BAADER
Abstract.
We prove that the links associated with positive elements of theoriented subgroup of the Thompson group are positive and alternating.
In memory of Vaughan F. R. Jones Introduction
The Thompson group F (along with its brothers T and V ) was introduced byR. Thompson in the sixties and has received a great deal of attention. Indeed, sev-eral equivalent definitions appeared in the literature, for instance as a subgroup ofpiecewise linear homeomorphisms of [0 , , as a diagram group, as pairs of planarrooted binary trees, and as strand diagrams. Motivated by the study of subfactors,Vaughan Jones started a new fascinating research program centred on the unitaryrepresentations of the Thompson groups. In particular, Jones’ recent work on therepresentation theory of Thompson’s group F gave rise to a combinatorial modelfor links, where elements of F define links in a similar way as elements of the braidgroups [13]. Unfortunately, the links arising from F do not admit a natural orienta-tion. For this reason, Jones introduced the so-called oriented subgroup (cid:126)F . The linksassociated with the elements of (cid:126)F come with a natural orientation. The Thompsongroups are as good knot constructors as the braid groups. In fact, all unoriented andoriented links can be produced by means of elements of F and (cid:126)F , respectively [13, 1].The purpose of this paper is to show that the notions of positivity for elements of (cid:126)F and for links are compatible, in the following sense. Theorem 1.
For any g ∈ (cid:126)F + , the oriented link (cid:126) L ( g ) admits a positive alternatingdiagram. Links admitting a positive alternating diagram are called special alternating. Pos-itive elements of the oriented group (cid:126)F admit a description by finite -valent rootedplane trees T . The associated links (cid:126) L ( T ) are defined diagrammatically via a binarytransform α ( T ) of the original tree. This is illustrated in the first two figures, anddefined in the next section. Positive oriented Thompson links are complicated inthat their defining diagrams are highly non-minimal. In particular, there tend to bea lot of unknotted components. As we will see, removing these trivial components,together with another type of simplification, results in special alternating link dia-grams whose crossing number is bounded above by the number of right leaves of the a r X i v : . [ m a t h . G T ] J a n VALERIANO AIELLO AND SEBASTIAN BAADER plane tree T . This is illustrated at the bottom of Figure 2, where the resulting linkis the positive twist knot . Figure 1.
The -regular rooted tree T and its binary transform α ( T ) . Figure 2.
Positive oriented Thompson link (cid:126) L ( T ) and its knottedcomponent . 2. Preliminaries and notation
In this section we review the basic definitions and properties of the Thompsongroup F , the oriented Thompson group (cid:126)F , the Brown-Thompson group F , and OSITIVE ORIENTED THOMPSON LINKS 3
Jones’s construction of knots from elements of Thompson group F . For furtherinformation we refer to [10, 8, 9, 13, 15].The Thompson group admits the following infinite presentation F = (cid:104) x , x , . . . | x n x k = x k x n +1 ∀ k < n (cid:105) . The monoid generated by x , x , . . . is denoted by F + and its elements are said to bepositive. In this paper we will make use of a graphical description of the elementsof F . Every element of F can be described by a pair of rooted planar binary trees ( T + , T − ) with the same number of leaves [10]. We draw such pairs of trees in theplane, with on tree upside down on top of the other and the leaves sitting on thenatural numbers of the x -axis (see Figure 3 for the generators of F ). Two pairsof trees are equivalent when they differ by a pair of opposing carets, see Figure 4.This equivalence relation allows to define the multiplication in F by the formula ( T + , T ) · ( T, T − ) := ( T + , T − ) . The trivial element is represented by any pair ( T, T ) and ( T + , T − ) − = ( T − , T + ) . x = x = x = . . . Figure 3.
The generators of F . Figure 4.
A pair of opposing carets and a pair of trees equivalent to x .The shift homomorphism ϕ : F → F is defined graphically as ϕ : (cid:55)→ g g This homomorphism maps x i to x i +1 for all i ≥ . VALERIANO AIELLO AND SEBASTIAN BAADER + = − = Figure 5.
A positive and a negative crossing.We now recall Jones’s construction [13, 15] of knots and links from elements of F ,which we illustrate with x as an example. The idea is to construct a Tait diagram Γ( T + , T − ) from a pair of trees ( T + , T − ) . We put the vertices of Γ( T + , T − ) on the halfintegers. For x x these points are (1 / , , (3 / , , (5 / , , (7 / , . The edges of Γ( T + , T − ) pass transversally through the edges of the top tree sloping up from leftto right (we call them West-North edges, or simply WN = | ) and the edges of thebottom tree sloping down from left to right (we refer to them by West-South edges,or just WS = | ). x x = Γ( x x ) = As shown in [13, Lemma 4.1.4] there is a bijection between the graphs of the form Γ( T + , T − ) and the pairs of trees ( T + , T − ) . We denote by Γ + ( T + ) and Γ − ( T − ) thesubgraphs of Γ( T + , T − ) contained in the upper and lower-half plane, respectively.Sometimes, to ease the notation we will use the symbols Γ + and Γ − . Since a Taitdiagram is a signed graph, we say that the edges of Γ + (resp. Γ − ) are positive(resp. negative). This means that in checkerboard shading of the correspondinglink diagram, the crossings corresponding to Γ + and Γ − are positive and negative,respectively (see Figure 5).In order to obtain a knot diagram we need two further steps. First we draw themedial graph M (Γ( T + , T − )) of Γ( T + , T − ) . In general, given a connected plane graph G , the vertices of its medial graph M ( G ) sit on every edge of G and an edge of M ( G ) connect two vertices if they are on adjacent edges of the same face. Belowwe will provide an example in our context. Now all the vertices of M (Γ( T + , T − )) have degree and to obtain a knot/link diagram we need to turn the vertices intocrossings. For the vertices in the upper-half plane we use the crossing / , while for OSITIVE ORIENTED THOMPSON LINKS 5 those in the lower-half plane we use . Here are M (Γ( x x )) and L ( x x ) M (Γ( T + , T − )) = L ( T + , T − ) = So far we have obtained an unoriented knot/link. In general the link diagramsobtained from elements of F do not admit a natural orientation. However, there isa natural orientation when the group element is in the oriented Thompson group (cid:126)F ,whose definition we now recall. Shade the the link diagram L ( T + , T − ) in black andwhite (we adopt the convention that the colour of the unbounded region is white).This yields a surface in R whose boundary is the link L ( T + , T − ) (see [13, Section5.3.2]). The oriented Thompson group (cid:126)F can be defined as [13] (cid:126)F := { ( T + , T − ) ∈ F | Γ( T + , T − ) is bipartite } . Equivalently, the elements of (cid:126)F have Tait diagram -colorable. We denote the coloursby { + , −} . We recall that if the Tait graph is -colorable, then there are exactly twocolorings. By convention we choose the one in which the leftmost vertex is assignedthe colour + . We denote by (cid:126)F + the monoid (cid:126)F ∩ F + .By construction the vertices of the graph Γ( T + , T − ) sit in the black regions andeach one has been assigned with a colour + or − . These colours determine anorientation of the surface and of the boundary ( + means that the region is positivelyoriented). It can be easily seen that the graph Γ( x x ) is bipartite and thus x x isin (cid:126)F (this element is actually one of the three natural generators of (cid:126)F ). Here is theoriented link associated with x x (cid:126) L ( T + , T − ) = + − + − The construction of the underlying unoriented links from elements of the Thompsongroup can also be obtained in the following equivalent way, [13, 15]. Starting froma tree diagram in F , first we turn all the -valent vertices into -valent and join thetwo roots of the trees, then we turn all the -valent vertices into crossings (see Figure6). We exemplify this procedure with x x VALERIANO AIELLO AND SEBASTIAN BAADER (cid:55)→ (cid:55)→ (cid:55)→
Figure 6.
The rules needed for obtaining L ( g ) . (cid:55)→ (cid:55)→ The Tait diagram of the link diagram obtained in this way is exactly the one de-scribed with the previous procedure. In passing, we mention that by means of planaralgebras [12] and this construction of knots, several unitary representation of boththe Thompson group and the oriented Thompson group related to notable knot andgraph invariants were defined [13, 14, 5, 2, 3, 4] and investigated [6, 16, 7].We recall that the bottom tree of a positive element in both F and the corre-sponding graph Γ − have the following form T − = . . . Γ − = . . . By convention the coloring of Γ − is + − + − + − . . . . Since the bottom tree ofa positive element is always the same, sometimes we will use the notation (cid:126) L ( T + ) ,instead of (cid:126) L ( T + , T − ) .It is well known that every element of the braid group may be expressed as theproduct of a positive braid and the inverse of a positive braid. A similar result fororiented Thompson group was proved by Ren [17]: Proposition 1.
For any g ∈ (cid:126)F , there exists g , g ∈ (cid:126)F + such that g = g g − . An oriented knot/link is positive if it admits a knot diagram where all the crossingsare positive, see Figure 7.The Brown-Thompson group F consists of pairs of rooted planar ternary treeswith the same number of leaves, [9]. The positive elements of F are those whose OSITIVE ORIENTED THOMPSON LINKS 7 + = − = Figure 7.
A positive and a negative crossing. (cid:55)→
Figure 8.
The isomorphism α : F → (cid:126)F .bottom tree is of the form T (cid:48)− = . . . The monoid consisting of positive elements in F is denoted by F , + . Since thebottom tree of a positive element (either in F and F ) is always of the same form,in the sequel we will only draw the top tree.In [11] it was proved that (cid:126)F is isomorphic with the Brown-Thompson group F .Later a graphical interpretation of this isomorphism was provided by Ren in [17]: inevery ternary tree, the -valent vertices are replaced by a suitable binary tree with leaves (see Figure 8). We will use this isomorphism in the next section to studythe positive oriented Thompson knots. Note that the trees of type T (cid:48)− are mappedto those of type T . Therefore, by its very definition α ( F , + ) is contained in F + ∩ (cid:126)F .Here follow some examples of positive oriented Thompson knots: (up to disjointunion with unknots) the trefoil, (3 + 2 n ) twist knot, the knot, the granny knot,and the oriented boundary of an n -times twisted annulus. Example 1.
The trefoil knot (up to disjoint union with unknots) may be obtainedfrom g := x x x x x x ∈ (cid:126)F + . Here is the top tree of g VALERIANO AIELLO AND SEBASTIAN BAADER and this is the corresponding Tait diagram Γ( g ) = Example 2.
The (3 + 2 n ) twist knot (up to disjoint union with unknots) is obtainedwith the following element in (cid:126)F + with top tree t n t n = . . . n + 1 Example 3.
The knot (up to disjoint union with unknots) can be obtained from g := x x x x x x x x x x x x x x ∈ (cid:126)F + with the following top tree t n = OSITIVE ORIENTED THOMPSON LINKS 9 and this is the corresponding Tait diagram Γ( g ) = Example 4.
The granny knot (up to disjoint union with unknots) can be obtainedfrom g := x x x x x x x x x x x x ∈ (cid:126)F + . Below is the top tree of g and this is the corresponding Tait diagram Γ( g ) = Example 5.
The oriented boundary of an n -times twisted annulus (up to disjointunion with unknots) is obtained with the following element in (cid:126)F + with top tree t n t n = . . . n ... type I ↔ ...... ... ± ∓ type IIa ↔ ... ... ±∓ ... ... type IIb ↔ ... ... Figure 9.
A Reidemeister move of type I allows to add (or remove)a 1-valent vertex and its edge. When there is a 2-valent vertex whoseedges have opposite signs, they may be contracted as shown in themove of type IIa. Two parallel edges with opposite signs may beadded (or removed) by means of a move of type IIb.
This is the Tait diagram whose associated link is T (2 , g ) = In Figure 9 we display the Reidemeister moves of type I and II in the language ofTait diagrams. They will come in handy in the next section.We conclude the preliminaries with a lemma concerning the structure of the toptree of elements in (cid:126)F + . It is a consequence of the proof of [17, Theorem 5.5] and forthis reason we only give a sketch of its proof. Lemma 1.
With the notations of the previous section, it holds α ( F ) ∩ (cid:126)F + = α ( F , + ) .Proof. The inclusion α ( F , + ) ⊂ α ( F ) ∩ (cid:126)F + is obvious. The converse inclusion canbe proved by induction on the number of leaves in the trees and by showing that thetop tree T + always contains the following subtree (the leaves of this subtree are asubset of the leaves of T + ) OSITIVE ORIENTED THOMPSON LINKS 11
If the leaves of the above tree are the rightmost leaves of T + , then by cancelling twopairs of opposing carets we are done. Otherwise multiply g ∈ (cid:126)F + by α ( x i x i +1 ) − (where i is a suitable non-negative integer). The element gα ( x i x i +1 ) − has less leavesthan g . Therefore, by induction gα ( x i x i +1 ) − = α ( g (cid:48) ) and we are done. (cid:3) Positive alternation of (cid:126) L ( (cid:126)F + ) In order to prove Theorem 1, we need to transform the link diagrams of elementsin (cid:126)F + into special alternating ones. Let g ∈ (cid:126)F + be represented by a pair of trees ( T + , T − ) , where T − is the standard bottom tree. By construction, the link diagram (cid:126) L ( g ) is a union of two tangles A and B , situated above and below the x -axis, respec-tively. Both tangles are alternating and have the same number of crossings n − ,where n is the number of leaves of the two trees T + , T − . Moreover, all the crossingsof A are positive, while all the crossings of B are negative, since the Tait graph Γ( T + , T − ) is bipartite. Therefore, in order to obtain a special alternating diagram,we need to remove all the crossings of the bottom tangle B . There is one negativecrossing attached to every string coming out of a leaf of T + , except for the leaf at thevery right of the tree, compare Figure 2. We will remove all these negative crossingssimultaneously, using two types of local moves. Recall that the upper tree can beinterpreted as the binary transform of a -valent plane tree, by Lemma 1. The cross-ings of B attached to left or middle vertices of the tree T + can be removed by singleReidemeister moves of type two, as shown in Figures 10 and 11. The small boxes inthese figures stand for an arbitrary -valent subtree. By these reduction moves, wealso remove one of the crossings of the upper tangle A . Figure 10.
Left leaf.The crossings of B attached to right vertices of the tree T + can be removed by adetour move, as shown in Figure 12. The precise effect of this move is removing anegative crossing, and replacing one of the crossings of A by another positive crossing.This can be seen by analysing the auxiliary orientations in the figure (in fact, thereare two possiblities for the local orientations; however, the actual choice has no effecton the signs of the crossings). We record two important features about these localmoves:(1) they can be preformed independently,(2) they preserve the alternation of the upper tangle A . Figure 11.
Middle leaf.In particular, we end up with a special alternating diagram for the link (cid:126) L ( g ) . Thisconcludes the proof of Theorem 1.The reader is invited to apply the above procedure to the top diagram of Figure 2.The resulting special alternating diagram is drawn at the bottom of the same figure,with the trivial components removed. Remark 1.
The number of crossings of the final diagram is bounded above by thenumber of right leaves of the upper tree T + . Figure 12.
Right leaf.4.
Unknotting positive oriented Thompson links
The next theorem provides an upper bound for the unknotting number of (cid:126) L ( g ) when g is in (cid:126)F + . We will use -valent plane trees in order to describe elements of (cid:126)F + , as in the proof of Theorem 1. As we will see, these trees can be reduced to anempty tree by using a set of seven moves as shown in Figure 13. Theorem 2.
For any g ∈ (cid:126)F + , the unknotting number is at most equal to the numberof applications of 6-move depicted in Figure 13.Proof. Let ( T + , T − ) be a pair of ternary trees in F . Thanks to Lemma 1 we canmake an induction on the number n of leaves. When n = 3 , the element ( T + , T − ) isthe trivial element of F and the corresponding link is trivial. Now there are seven OSITIVE ORIENTED THOMPSON LINKS 13
1) 2) 3)4) 5) 6)7)
Figure 13.
Seven reduction moves for the elements in F , + = α − ( (cid:126)F + ) . All these moves except the sixth, do not affect the cor-responding knot (only some unknots are lost in the application ofthese moves). In the sixth move a positive crossing is turned into anegative one.cases to deal with. Indeed, an easy inductive argument shows that T + contains oneof the subtrees depicted in Figure 13 (the leaves of this tree are a subset of the leavesof T + ) The edges in red in the above figure will be erased.As we shall see, in all the cases, but case 6), we simply apply Reidemeister movesof type I and II, and (possibly) remove unknots. Only in case 6) we need to turn apositive crossing into negative.Here follows the subtree of case 1) transformed under the map α , the correspondingTait diagram, an "equivalent" Tait diagram along with the corresponding binary andternary trees.Here are the analogous graphs for case 2)In the third case we only need Reidemeister moves of type II as shown below These are the analogous graphs for the fourth caseHere is case 5)The sixth case is more complicated. First we simplify the Tait diagramThere are two subcases depending on the color of the leftmost vertex. Suppose thatthe color is + , then we have the following Tait diagram and the link + − + − After turning the leftmost crossing into a negative crossing we get the following link,Tait diagrams, tree + − + − + − Similarly when the color is − , then we have the following Tait diagram, the link,Tait diagrams and tree − + − + + − + − + − Finally, we take care of case 7)
OSITIVE ORIENTED THOMPSON LINKS 15 (cid:3)
Remark 2.
The above bound is optimal. Indeed, let g be x x x x , then (cid:126) L ( g ) is theHopf link (up to disjoint union with unknots). The link (cid:126) L ( gϕ ( g )) is a chain linkwith connected components. More generally, (cid:126) L ( (cid:81) n − i =0 ϕ i ( g )) is a chain link with n connected components, see Figure 14 (here we are using the notation (cid:81) ni =1 g i := g · · · g n ). It is easy to see that the unknotting number and the number of 6-movesneeded are both equal to n . T n = T n − Figure 14.
The element (cid:81) n − i =0 ϕ i ( g ) is represented by the aboverecursive relation, where T is the empty tree. The correspondinglink is a chain with n connected components.In the proof of Theorem 2 we saw that elements g ∈ (cid:126)F + can be simplified by turningpositive crossings into negative crossings. We end this paper with the observationthat it is also possible to reduce the "size" of g by smoothing suitable positivecrossings. As before we use an inductive argument on the number of leaves of anelement ( T + , T − ) in F , + . By the previous discussion, one of the cases depictedin Figure 13 occurs. In all cases but , we may reduce the number of leaves withthe only effect of losing some distant unknots. In the sixth we changed a positivecrossing into a negative one. In this case, we claim that it also possible to reduce thesize of g by smoothing a crossing. As in the previous proof, there are two subcasesdepending on the color of the leftmost vertex. Suppose that the color is + , then wehave the following Tait diagram and the link + − + − If we smooth the leftmost crossing, after applying some Reidemeister moves, we getthe following Tait diagram + − + − (cid:48) ) Figure 15.
An additional move of type 6.Similarly when the color is − , we have the following knot and Tait diagrams − + − + and after smoothing the leftmost crossing, we get the following Tait diagram + − + − In both cases we get a new element ( T (cid:48) + , T (cid:48)− ) , where T (cid:48) + has n − leaves. Acknowledgements
The authors acknowledge the support by the Swiss National Science foundationthrough the SNF project no. 178756 (Fibred links, L-space covers and algorithmicknot theory).
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Valeriano Aiello, Mathematisches Institut, Universität Bern, Alpeneggstrasse 22,3012 Bern, Switzerland
Email address : [email protected] Sebastian Baader, Mathematisches Institut, Universität Bern, Sidlerstrasse 5,3012 Bern, Switzerland
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