Maximal states of alternating links with applications Xian’an Jin School of Mathematical Sciences, Xiamen University, P. R. China SJTU, May 10-13, 2013
This talk is a simplified version of the following manuscript: Jin Xian’an, Ge Jun, Xiao-Sheng Cheng, Maximal states of al- ternating links with applications, submitted. It contains 3 parts: Part I. The genus of a link Part II. The circle number of maximal states Part III. The coincidence and applications 1
Part I. The genus of a link Knots and links A (classical) knot is a simple closed curve in R 3 . A link is the disjoint union of a finite number of knots, usually interlocked with each other; each knot is called a component of the link. A knot is a link with one component. Let L be a link. We denote by µ ( L ) the number of components of L . 2
Examples: several simplest knots and links. 3
Notes. • Links in R 3 ⊂ S 3 are called classical links, distinguishing from links in other 3-dimensional manifolds. • To avoid some pathological examples, say, some infinitely knotted loops, we demand that the closed curve forming a knot is polygonal. In other words, we only consider so-called tame links not wild links. • We shall always draw links smoothly only for the aesthetic purpose. 4
Link diagrams Although (tame) links live in R 3 , we usually represent them by link diagrams : the projection of the link into R 2 with under- passing curves indicated by breaks. There are some technical requirements for the projection such that the projection to be a diagram, and the existence of such projection for a tame link is non-trivial mathematically. We do not talk about details here. 5
Examples: Standard diagrams of trefoil knot, Listing knot, Hopf link and Borromean Rings. 6
An oriented link ⃗ L is a link L with each of its components specified an orientation. Two (oriented) links are equivalent or ambient isotopic if in- tuitively one can be deformed to the other (with orientations respected). We do not talk about the mathematically rigorous definition of ambient isotopy here. 7
Examples: The trivial knot (upper) and the Listing knot (lower). 8
A central question in knot theory is to develop various methods to judge that if two (oriented) links are equivalent. A link invariant is a function from the set of all (oriented) links to any other set such that the function does not change as the (oriented) link is deformed. In other words, a link invariant always assigns the same value to equivalent (oriented) links (although different links may have the same link invariant), a pair of (oriented) links with different invariants must be not equivalent. 9
There are many link invariants. In this talk, we are only interested in the genus invariant of links. Now we recall the definition of the genus of a link. The genus of a link Let ⃗ L be an oriented link. An orientable, connected surface that has ⃗ L as its boundary is called a Seifert surface of ⃗ L . Given a diagram ⃗ D of ⃗ L , using Seifert algorithm in 1935, we can obtain a Seifert surface called projection surface formed by disks and twisted bands. 10
Example: The Listing knot and its projection surface. The genus of an oriented link is the minimum genus of all Seifert surfaces of the oriented link. The genus of an unoriented link is the minimum taken over all possible choices of orientation. We denote the genus of an (unoriented) link L by g ( L ). 11
A link diagram is said to be alternating if it has crossings that alternate between over and under as one travels around each component in a fixed direction. A link is said to be alternating if it admits an alternating link diagram. Theorem A. (Murasugi & Crowell, 1958) Let ⃗ L be an alternating oriented link. Let ⃗ D be an alternating link diagram of ⃗ L . By applying Seifert’s algorithm to ⃗ D one obtains a Seifert surface of minimal genus of ⃗ L , i.e. the projection surface is with the minimal genus among all Seifert surfaces. 12
Let F be a connected orientable surface formed from d disks and b bands (is now called a cyclic graph or an orientable ribbon graph). Then the Euler Characteristic of F χ ( F ) = d − b. (1) And hence, the genus of F g ( F ) = 2 − d + b − bc ( F ) (2) , 2 where bc ( F ) is the number of boundary components of F . d ← → # circles → # crossings b ← bc ( F ) ← → # components It is easy to see that 13
Theorem B. Let D be an alternating link diagram. Let L be the link D represents. Let ¯ s max ( D ) be the maximal number of circles by applying Seifert’s algorithm to all orientations of D . Then g ( L ) = 2 − ¯ s max ( D ) + c ( D ) − µ ( D ) (3) , 2 where c ( D ) and µ ( D ) are the number of crossings and link com- ponents of D . Hence, if the link L has µ ( L ) components, we need to consider 2 µ ( L ) different orientations, compute each number of circles and take the maximum over them. The main difficulty lies in that there are too many orientations where µ ( L ) is large. 14
Part II. The circle number of maximal s- tates 15
Let D be an unoriented link diagram. A state S of D is a labeling each crossing of D by either A or B . Making the corresponding split B A A B for each crossing gives a number of disjoint embedded closed circles, called state circles, for S . We write s for the number of state circles of the state S . There are two extreme states S A and S B corresponding to all A -splits and B -splits, respectively, and accordingly write s A and s B for the numbers of state circles of S A and S B . We call the state which possesses maximal number of state circles to be a maximal state. 16
Example. Hopf link D B B B B A A A A A A A A B B B B B B B B A A A A A A A A B B B B Thus, s max ( D ) = s A ( D ) = s B ( D ) = 2. 17
Link diagrams from plane graphs via medial construction We first define the medial graph of a plane graph. The medial graph M ( G ) of a plane graph G is defined as follows. If G is trivial (i.e. an isolated vertex having no edges), M ( G ) is a simple closed curve surrounding the vertex. If G is a connected non-trivial plane graph, M ( G ) is a 4-regular plane graph obtained by inserting a vertex on every edge of G , and joining two new vertices by an edge lying in a region of G if the two vertices are on adjacent edges of the region. If G is not connected, M ( G ) is defined to be the disjoint union of the medial graphs of its connected components. 18
Example. A plane graph (black) and its medial graph (red). Given a plane graph G , we first draw its medial graph M ( G ), then turn M ( G ) into a link diagram D ( G ) by turning the vertices of M ( G ) into crossings as shown in the following example. 19
Example. The above procession can be reversed via the checker-board col- oring, hence an 1-1 correspondence between plane graphs and alternating link diagrams (up to mirrors) is established. This correspondence was actually discovered more than one hundred years ago which was once used to knot tabulation in the early time of knot theory. 20
Let G be a plane graph with v ( G ) vertices, e ( G ) edges, f ( G ) faces and k ( G ) connected components. Then s A ( G ) := s A ( D ( G )) = f ( G ) + k ( G ) − 1 , (4) s B ( G ) := s B ( D ( G )) = v ( G ) . (5) Let G be a plane graph. We write s max ( G ) for the number of state circles of the maximal state of D ( G ). After trying some graphs, one may guess that s max ( G ) = max { s A ( G ) , s B ( G ) } . (6) 21
This guess is not true. For example In fact, let G n be the plane graph obtained from the n -cycle C n ( n ≥ 4) by adding n − 2 parallel edges to a specific edge of C n , then f ( G n ) = v ( G n ) = n . Let S be the state of D ( G ( n )), crossings corresponding to n − 1 parallel edges labeled by A , others labeled by B . Then s = 2 n − 3 > n . This means that s max ( G ) − max { s A ( G ) , s B ( G ) } can be large ar- bitrarily. 22
More families of plane graphs without parallel edges or vertices of degree 2 are also found. In the following, we suppose that G is a connected plane graph and shall discuss when s max ( G ) = s A ( G ) = f ( G ). We obtain that Lemma C. If G is a 3-edge connected even plane graph and s is the number of state circles of any state S of D ( G ). Then s ≤ s A . In other words, s max ( G ) = s A ( G ) = f ( G ). We omit the proof of the Lemma C. 23
Remarks. 1. The following example implies that the 3-edge connectedness in Lemma C is necessary. 24
2. The following example implies that there are graphs which are not 3-edge connected but S A possesses the maximal number of circles. 25
Part III. Coincidence and applications 26
The coincidence Let D be a link diagram. Let ⃗ D be the diagram D with a fixed orientation. Then the orientation will induce a state of D called a projection state. Example: The oriented Listing knot and its corresponding pro- jection state. A A B B 27
Hence, we always have ¯ s max ( D ) ≤ s max ( D ) . (7) Lemma D. Let G be an even plane graph. Then D ( G ) has an orientation whose corresponding projection state is exactly the all- A state. Proof. It is well known that the dual graph of an even plane graph is bipartite. Hence regions (i.e. faces) of G can be shaded in a checkerboard fashion. 28
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