Intersection Graphs for String Links

Transcription

1 Dgtal Loyola Marymount Unversty and Loyola Law School Mathematcs Faculty Works Mathematcs Intersecton Graphs for Strng Lnks Blake Mellor Loyola Marymount Unversty, Repostory Ctaton Mellor, Blake, "Intersecton Graphs for Strng Lnks" (2006). Mathematcs Faculty Works Recommended Ctaton Mellor, B., 2006: Intersecton Graphs for Strng Lnks. J. Knot Theory Ramf., 15.1, 53-72, arxv:math/ Ths Artcle - post-prnt s brought to you for free and open access by the Mathematcs at Dgtal Loyola Marymount Unversty and Loyola Law School. It has been accepted for ncluson n Mathematcs Faculty Works by an authorzed admnstrator of Dgtal Commons@Loyola Marymount Unversty and Loyola Law School. For more nformaton, please contact dgtalcommons@lmu.edu.

2 Intersecton Graphs for Strng Lnks arxv:math/ v2 [math.gt] 7 Jan 2005 Blake Mellor Mathematcs Department Loyola Marymount Unversty Los Angeles, CA bmellor@lmu.edu Abstract We extend the noton of ntersecton graphs for chord dagrams n the theory of fnte type knot nvarants to chord dagrams for strng lnks. We use our defnton to develop weght systems for strng lnks va the adjacency matrx of the ntersecton graphs, and show that these weght systems are related to the weght systems nduced by the Conway and Homfly polynomals. Contents 1 Introducton 1 2 Prelmnares Knots, Lnks and Strng Lnks Fnte Type Invarants Intersecton Graphs for Strng Lnks Defnton Graph balgebra for Strng Lnks term relatons for Strng Lnks Chord dagrams for Strng Lnks modulo 2-term relatons Adjacency Matrces for Intersecton Graphs Relatons to the Conway and Homfly polynomals Further Questons and Problems 15 1 Introducton The theory of fnte type nvarants allows us to nterpret many mportant knot and lnk nvarants n purely combnatoral terms, as functonals on spaces of chord dagrams. For knots, there s an obvous ntersecton graph assocated wth these dagrams, frst studed (n ths context) by Chmutov, Duzhn and Lando [6]. In many cases, these graphs contan all the relevant nformaton for the functonals comng from knot nvarants [6, 12]. Some of the most mportant knot nvarants, such as the Conway, Jones, Homfly and Kauffman polynomals, can be nterpreted n terms of these ntersecton graphs [2, 13]. However, t s not obvous how to extend the dea of the ntersecton graph to fnte type nvarants of other objects, such as brads, lnks and strng lnks. The goal of ths paper s to ntroduce a reasonable defnton of the ntersecton graph for chord dagrams assocated wth strng lnks. As evdence that ths s the rght defnton, we use these ntersecton graphs to construct weght systems for strng lnks, and show 1

3 that these weght systems are related to (though weaker than) the weght systems arsng from the Conway and Homfly polynomals. In future work we wll show that n some cases the ntersecton graph contans all the relevant nformaton n the strng lnk chord dagram [14, 15], and that t can be used to gve a new nterpretaton of Mlnor s homotopy nvarants [16]. In secton 2 we wll revew the necessary background, ncludng fnte type nvarants, chord dagrams, ntersecton graphs and Lando s graph balgebra [9]. In secton 3 we wll defne ntersecton graphs for strng lnks, and provde the space of graphs wth a balgebra structure smlar to Lando s. In secton 4 we look at chord dagrams modulo the 2-term relatons ntroduced by Bar-Natan and Garoufaldes [2], and use the adjacency matrces of ther ntersecton graphs to construct weght systems related to those arsng from the Conway and Homfly polynomals. Fnally, n secton 5 we pose some questons for further research. Acknowledgement: The author thanks Loyola Marymount Unversty for supportng ths work va a Summer Research Grant n Prelmnares 2.1 Knots, Lnks and Strng Lnks An (orented) knot s an embeddng of the (orented) crcle S 1 nto the 3-sphere S 3. A knot nvarant s a map from these embeddngs to some set whch s nvarant under sotopy of the embeddng. We wll also consder nvarants of regular sotopy, where the sotopy preserves the framng of the knot (.e. a chosen secton of the normal bundle of the knot n S 3 ). A lnk wth k components s smply an embeddng of the dsjont unon of k copes of S 1 nto S 3 ; each component s a knot. A strng lnk can be defned as follows: Defnton 1 (Habegger and Ln [7]) Let D be the unt dsk n the plane and let I = [0,1] be the unt nterval. Choose k ponts p 1,..., p k n the nteror of D, algned n order along the the x-axs. A strng lnk σ of k components s a smooth proper mbeddng of k dsjont copes of I nto D I: σ : k I D I =1 such that σ I (0) = p 0 and σ I (1) = p 1. The mage of I s called the th strng of the strng lnk σ. Lnk and strng lnk nvarants are defned n the same way as for knots. Note that any strng lnk can be closed up to a lnk n a unque way by jonng the top and bottom of each component by an arc whch les outsde of D I (and whch s unlnked wth the other such arcs). So any lnk nvarant gves rse to a strng lnk nvarant by evaluatng t on the closure of the strng lnk. 2.2 Fnte Type Invarants Our treatment of fnte type nvarants wll follow the combnatoral approach of Brman and Ln [3]. We wll gve a bref overvew of ths combnatoral theory, and ts natural extensons to lnks and strng lnks. For more detals, see Bar-Natan [1]. We frst note that we can extend any lnk nvarant to an nvarant of sngular lnks, where a sngular lnk s an mmerson of a dsjont unon of copes of S 1 n 3-space whch s an embeddng except for a fnte number of solated double ponts. Gven a lnk nvarant v, we extend t va the relaton: An nvarant v of sngular lnks s then sad to be of fnte type, specfcally of type n, f v s zero on any lnk wth more than n double ponts (where n s a fnte nonnegatve nteger). We denote by V n the vector space 2

4 (1-term relaton) (4-term relaton) = = 0 Fgure 1: The 1-term and 4-term relatons. No other chords have endponts on the arcs shown. In the 4-term relatons, all other chords of the four dagrams are the same. over C generated by (framng-ndependent) fnte type nvarants of type n. We can completely understand the space of fnte type nvarants by understandng all of the vector spaces V n /V n 1. An element of ths vector space s completely determned by ts behavor on lnks wth exactly n sngular ponts. In addton, snce such an element s zero on lnks wth more than n sngular ponts, any other (non-sngular) crossng of the knot can be changed wthout affectng the value of the nvarant. Ths means that elements of V n /V n 1 can be vewed as functonals on the space of chord dagrams: Defnton 2 A chord dagram of degree n wth k components s a dsjont unon of k orented crcles, together wth n chords (lne segments wth endponts on the crcles), such that all of the 2n endponts of the chords are dstnct. The crcles represent a lnk of k components and the endponts of a chord represent 2 ponts dentfed by the mmerson of ths lnk nto 3-space. The dagram s determned by the orders of the endponts on each component. For strng lnks, chord dagrams nvolve orented lne segments rather than crcles. Functonals on the space of chord dagrams whch are derved from fnte type lnk nvarants wll satsfy certan relatons: Defnton 3 A weght system of degree n s a lnear functonal W on the space of chord dagrams of degree n (wth values n an assocatve commutatve rng K wth unty) whch satsfes the 1-term and 4-term relatons, shown n Fgure 1. For knots, the three arcs all belong to the same crcle; for lnks (resp. strng lnks), they may belong to the same or dfferent crcles (resp. lne segments). The natural map from elements of V n /V n 1 to functonals on chord dagrams s a homomorphsm nto the space of weght systems [1, 3, 20, 21]. Kontsevch proved the much more dffcult fact that these spaces are somorphc [1, 8] (the nverse map s the famous Kontsevch ntegral). For convenence, we take the dual approach, and smply study the space of chord dagrams of degree n modulo the 1-term and 4-term relatons. The 1-term relaton s occasonally referred to as the framng-ndependence relaton, because t arses from the framng-ndependence of the nvarants n V n (essentally, from the frst Redemester move). Snce most of the nterestng structure of the vector spaces arses from the 4-term relaton, t s common to look at the more general settng of nvarants of regular sotopy, and consder the vector space A k n of chord dagrams of degree n on lnks wth k components modulo the 4-term relaton alone. Lnear functonals on A k n are called regular weght systems of degree n. Smlarly, we defne the vector space Bn k for chord dagrams of degree n on strng lnks of k components. It s useful to combne all of these spaces nto graded modules A k = n 1 Ak n and Bk = n 1 Bk n va drect sum. For k = 1, we can defne a product on A 1 va connect sum [1]; however, ths does not extend to k > 1. But we can gve the module B k a balgebra (or Hopf algebra) structure for any k by defnng an approprate product and co-product: We defne the (noncommutatve) product D 1 D 2 of two chord dagrams D 1 and D 2 as the result of 3

5 placng D 2 on top of D 1 (jonng the components so the orentatons agree), as shown below: = D 2 D 1 D 2 D 1 We defne the co-product (D) of a chord dagram D as follows: (D) = J D J D J where J s a subset of the set of chords of D, D J wth all the chords not n J removed. s D wth all the chords n J removed, and D J s D ( ) = It s easy to check the compatblty condton (D 1 D 2 ) = (D 1 ) (D 2 ). 3 Intersecton Graphs for Strng Lnks For knots, there s a very natural noton of ntersecton among the chords of a chord dagram - two chords ntersect f ther endponts alternate around the boundng crcle. The ntersecton graph then has a vertex for each chord, and connects two vertces by an edge f the correspondng chords ntersect. The dffculty wth extendng ths dea to lnks and strng lnks s how to deal wth chords whch have ther endponts on dfferent components of the dagram - specfcally, f they each have one endpont on a gven component, and ther other endponts are on dfferent components. In what sense, f any, do these chords ntersect? 3.1 Defnton The essental value of the ntersecton graph for knots s that t can detect when the order of two endponts for dfferent chords along the boundng crcle s swtched (as happens n the 4-term relaton), snce ths changes the par of chords from ntersectng to non-ntersectng or vce-versa. To usefully extend the noton of ntersecton graphs to lnks or strng lnks, we need to retan ths ablty. For strng lnks, the exstence of a bottom and top for each component allows us to gve a lnear (rather than cyclc) orderng to the endponts of the chords on each component, and so the noton of one endpont beng below another s well-defned. Defnton 4 Let D be a strng lnk chord dagram wth k components (orented lne segments, colored from 1 to k) and n chords. The ntersecton graph Γ(D) s the labeled, drected multgraph such that: Γ(D) has a vertex for each chord of D. Each vertex s labeled by an unordered par {, j}, where and j are the labels of the components on whch the endponts of the chord le. 4

6 {,} {,} {,} = {,} {,} {,} = = j j {,} {j,k} {k,k} j k {j,k} j k {,} {j,k} Fgure 2: Examples of ntersecton graphs for strng lnks There s a drected edge from a vertex v 1 to a vertex v 2 for each par (e 1, e 2 ) where e 1 s an endpont of the chord assocated to v 1, e 2 s an endpont of the chord assocated to v 2, e 1 and e 2 le on the same component of D, and the orentaton of the component runs from e 1 to e 2 (so f the components are all orented upwards, e 1 s below e 2 ). We count these edges mod 2, meanng that f two vertces are connected by two drected edges wth the same drecton, they cancel each other. If two vertces are connected by a drected edge n each drecton, we wll smply connect them by an undrected edge. Examples of chord dagrams and ther assocated ntersecton graphs are gven n Fgure 2. Note that when the two chords have both endponts on the same component, our defnton of ntersecton graph corresponds to the usual ntersecton graph for knots. Our defnton also matches our ntuton n the case of chord dagrams of two components, as shown n Fgure 2. Note also that the total number of drected edges between a vertex v labeled {, j} and a vertex w labeled {l, m} s gven by the sum of the number of occurrences of n {l, m} and the number of occurrences of j n {l, m}. In partcular, f a vertex v has a label {, }, ths number wll be even (0, 2 or 4). Snce we count drected edges modulo 2, ths mples there s an (uncancelled) drected edge from v to another vertex w f and only f there s also an (uncancelled) drected edge from w to v. We wll say that labeled drected multgraphs whch have ths property are semsymmetrc. 5

7 Defnton 5 A drected multgraph G, wth each vertex labeled by a par, s semsymmetrc f for every vertex v labeled {,}, and any other vertex w, there s a drected edge from v to w f and only f there s a drected edge from w to v. 3.2 Graph balgebra for Strng Lnks Followng Lando s work for knots [9], we can defne a balgebra structure on the space of ntersecton graphs for strng lnks so that Γ becomes a balgebra homomorphsm from the space of chord dagrams for strng lnks to the space of ntersecton graphs. The key s to defne the analogue of the 4-term relaton for ntersecton graphs. Defnton 6 Consder the graded vector space (over C) of formal lnear combnatons of labeled semsymmetrc drected multgraphs, wth vertces labeled by unordered pars, 1, j k, graded by the number of vertces n the graphs. For any graph G and vertces A and B n V(G), wth labels and {,l} respectvely (j and l may be equal, or equal to ), we mpose on the vector space the relaton: G G AB G AB + G AB = 0 Here G AB s the result of complementng the edge AB n G (.e. addng a drected edge between the vertces n each drecton, and then cancellng mod 2 ). GAB s the result of changng the label on A to {j,l}, complementng the edge AB f l (and leavng t unchanged f = l), and addng a drected edge from A to C (respectvley C to A) for every vertex C n V(G) for whch G has a drected edge from B to C (respectvely C to B), and then cancellng mod 2. Fnally, G AB s the result of complementng the edge AB n G AB. Here s an example of such a relaton: {j, l} {j, l} A B A B A B A B {, j} {, l} - {, j} {j, l} {, l} - {, l} {, l} = 0 {j, l} {j, l} + {j, l} {, } {l, k} {, } {l, k} {, } {l, k} {, } {l, k} The balgebra F s defned as ths graded vector space, together wth a product and a coproduct. The product s a map : F F F, defned as follows. Gven graphs G 1 and G 2, G 1 G 2 s the dsjont unon of the graphs, together wth a drected edge from v 1 V (G 1 ) to v 2 V (G 2 ) for each color n the label for v 1 whch s also n the label for v 2. An example s shown below: G 1 = G 2 = {,} {j,l} {j,k} {j,k} G 1 G = 2 {,} {j,l} The coproduct s a map µ : F F F, defned as follows. For any graph G, and subset J V (G) of ts vertces, let G J denote the subgraph nduced by J. Then: µ(g) = G J G V (G)\J J V (G) 6

8 An example s shown below: µ( {j,k} ) = {j,k} + {j,k} 1 + {j,k} + 1 {j,k} Theorem 1 The product and coproduct defned above nduce the structure of a co-commutatve (but not commutatve) balgebra on the space F of labeled, drected multgraphs modulo the 4-term relaton. Proof: It s easy to check that F s a balgebra (.e. that µ(g 1 G 2 ) = µ(g 1 ) µ(g 2 )), and that t s co-commutatve, but not commutatve. We also need to check that the product and coproduct respect the 4-term relaton,.e. that the product of a graph and a 4-term relaton yelds a 4-term relaton, and that the coproduct of a 4-term relaton yelds a sum of 4-term relatons. To begn wth, consder a product (G G AB G AB + G AB ) H = G H G AB H G AB H+ G AB H. It s clear that G AB H = (G H) AB, snce complementng the edge AB s ndependent of any edges between G and H. Moreover, G AB H = G H AB, snce n each case there s an drected edge from a vertex v V (H) to A for each occurrence of j or l n the label of v (wth cancellaton mod 2). It s then clear that G AB H = G H AB, and so the product of the 4-term relaton and another graph s another 4-term relaton. For the coproduct, consder µ(g G AB G AB + G AB ). The sum n the coproduct splts nto two groups - terms where both A and B belong to ether J or ts complement, and terms where one of the two vertces s n J and the other s n ts complement. The frst group gves a sum of 4-term relatons, whle the terms of the second group already sum to zero n µ(g G AB ) and µ( G AB G AB ). Ths fnshes the proof of the theorem. It s now easy to check that Γ s a balgebra homomorphsm from the balgebra B k of chord dagrams for strng lnks wth k components modulo the 4-term relaton to the balgebra F k of drected, labeled graphs wth labels 1,..., k, snce the 4-term relaton for graphs was defned to mmc the 4-term relaton for the chord dagrams. As a result, any functonal on F k wll nduce, by composton wth Γ, a regular weght system. We call such a functonal a regular graph weght system. 4 2-term relatons for Strng Lnks Any partcular weght system wll satsfy relatons n addton to the 1-term and 4-term relatons, and t can be useful to look at weght systems whch le n the subspaces determned by these addtonal relatons. In partcular, Bar-Natan and Garoufaldes [2] noted that the weght system assocated wth the Conway polynomal for knots satsfes a set of 2-term relatons whch nduce the 4-term relatons. We can extend ths noton to defne a set of 2-term relatons for strng lnks, shown n Fgure 3. Clearly, these relatons - = 0 Fgure 3: The 2-term relatons for strng lnks mply that the weght system satsfes the 4-term relaton as well. As a result, the product and coproduct of secton 2.2 are stll well-defned. So we can gve the vector space of chord dagrams for strng lnks wth k components modulo the 2-term relatons the structure of a balgebra. We wll denote ths balgebra (and the underlyng vector space) by D k. There s a natural projecton from B k to D k. 7

9 Smlarly, we can look at the quotent space of F by the 2-term relatons G G AB = 0 Snce G AB = G AB, the 2-term relatons mply the 4-term relatons, and the quotent space s a balgebra E wth the product and coproduct nduced from F. Γ s then a balgebra homomorphsm from D k to E k. 4.1 Chord dagrams for Strng Lnks modulo 2-term relatons Bar-Natan and Garoufaldes [2] analyzed the space of chord dagrams for knots (equvalently, strng lnks of one component) modulo the 2-term relatons. They found that D 1 s generated (as a vector space) by (m 1, m 2 )-caravans of m 1 one-humped camels (solated chords whch ntersect no other chords) and m 2 two-humped camels (pars of chords whch ntersect each other, but no other chords). An example of such a caravan s shown n Fgure 4. Fgure 4: Example of an (m 1, m 2 )-caravan Our goal n ths secton s to fnd a smlar normal form for strng lnk chord dagrams wth more than one component; ths wll be useful n the remander of the paper. We wll prove the followng theorem: Theorem 2 Any connected strng lnk chord dagram s equvalent, modulo the 2-term relatons, to a chord dagram wth components numbered 1,..., n (possbly after renumberng) such that: Every chord ether has both endponts on component 1, or endponts on components and +1 for some. The chords wth both endponts on component 1 are arranged n a caravan. Moreover, all of ther endponts le below the endponts of any chords between components. There are at most 2 chords between components and +1, and f there are 2 they do not cross (so the endponts of one chord le above the endponts of the other on both components). On component, the endponts of chords connectng component to component +1 le below the endponts of chords connectng component to component -1. An example s shown n Fgure 5. caravan Fgure 5: Example of a lnk chord dagram n normal form 8

10 If a dagram s not connected, we smply put each connected component nto normal form. The proof proceeds n several steps. We wll descrbe an algorthm to put any connected chord dagram nto the normal form descrbed above va 2-term relatons. Step 1: We frst deal wth chords whch have both endponts on the same component. To begn wth, any par of ntersectng chords whch have all four endponts on the same component can be sld to the bottom of the component, away from any other chords, to form a two-humped camel as n [2]. An solated chord wth both endponts on the same component can also be sld down to the bottom, formng a one-humped camel. See Fgure 6 for examples of these moves. We are left wth chords wth both endponts on the same Fgure 6: Factorng out one- and two-humped camels component whch only ntersect chords whch le between two dfferent components. By sldng ths chord over one of the chords t ntersects, ts endponts are now on two dfferent components, as shown n Fgure 7. j j Fgure 7: A move from Step 1 Step 2: Our next step s to put every chord connectng two dfferent components on a dfferent horzontal level ; n partcular, we wll remove ntersectons between chords between the same two components. Say that we have n chords between components. Select a chord arbtrarly, whch we wll denote chord c. Wthout loss of generalty, assume c connects components 1 and 2. We wll move c to the top level. As we move t up, f we come to a chord wth an endpont on components 1 or 2 whch s above c we move that endpont below c va a 2-term relaton, as shown n Fgure 8. Ultmately, the endponts of c wll le above the endponts of c c Fgure 8: Movng an endpont below c any other chord. Durng ths process, some chords may have had both ther endponts moved to the same component - remove or modfy these chords as n Step 1 (notce that ths only reduces the number of chords 9

11 between components from n). We are left wth at most n 1 chords connectng components, all below c. So we can contnue the process nductvely, endng wth all chords between components horzontal, at dfferent levels (and wth a caravan at the bottom of each component). Step 3: Now we consder one component of the dagram, say component 1, and move all the chords wth an endpont on component 1 to a hgher level than all the other chords n the dagram (leavng chords wth both endponts on the same component at the bottom). We wll denote by an (,j)-chord a chord wth endponts on components and j. If a (1,)-chord les drectly below an (, j)-chord along component, we can use a 2-term relaton to move the (1,)-chord above the other chord, n the process transformng t nto a (1,j)-chord, as n Fgure 9. Snce there are only a fnte number of chords, we can successvely move all 1 j 1 j Fgure 9: Step 3 chords wth one endpont on component 1 above all other chords (whle the other endpont may move, one wll always reman on component 1). Step 4: Let c be the top chord at ths pont; wthout loss of generalty, c s a (1,2)-chord. Let k 1 be the number of (1,)-chords, and k 1 = k 1. Now move c down component 1, strppng other chords off of component 1 by 2-term relatons, transformng them from (1,)-chords to (2,)-chords, reversng the move from Step 2 shown n Fgure 8. Contnue ths untl c encounters another (1,2)-chord, d. Now move c back up as n Step 3 (see Fgure 9); when t s agan at the top, t wll be a (1,)-chord for some. Repeat the process for d, movng t up to just below c, and wth all the other (1,2)-chords. Ths reduces k 1 to k 12, though the chords may no longer be (1,2)-chords. We can relabel the components so that c s once agan a (1,2)-chord and repeat the process. Each repetton reduces k 1, untl we reach a pont where k 1 = k 12 (.e. all chords wth an endpont on component 1 are (1,2)-chords), and the (1,2)-chords le above all other chords n the dagram. Fnally, we can reduce k 12 to 1 or 2 by notcng that whenever k 12 3, we can factor out a 2-humped camel and reduce k 12 by 2, as n Fgure 10. j j j j Fgure 10: Factorng a two-humped camel from 3 parallel chords Step 5: We can now repeat Steps 3 and 4 for component 2, and then for each component n turn. We are left wth a dagram whch s almost n normal form - the fnal step s to move all of the caravans to component 1. We can move a caravan from the th component to the ( 1)th component by sldng t over a chord between the two components, as n Fgure 11. Any other chords between the two components can be sld over the caravan n turn by 2-term relatons, leavng the caravan below the chords connectng components 1 and 2. Contnung n ths way, the caravans can be moved to component 1, and fnally pushed to to bottom of component 1 as n step 1. Ths completes the proof of Theorem 2. 10

12 C C C j j j Fgure 11: Movng a caravan 4.2 Adjacency Matrces for Intersecton Graphs In ths secton we wll defne the adjacency matrx for an ntersecton graph, and show that ts rank and determnant (over Z 2 ) are regular graph weght systems. Defnton 7 Gven a drected graph G wth n vertces, {v 1,..., v n }, such that each vertex v has a label {a, b }, the adjacency matrx of G, or adj(g), s the n n matrx defned by: { 1 f there s a drected edge from v to v adj(g) j ( j) = j n G 0 otherwse { 1 f a b adj(g) = 0 f a = b These matrces can be vewed as blnear forms over Z 2, but unlke the case for knots [13], they are not generally symmetrc, so ther classfcaton s much more dffcult (and stll, to my knowledge, an open queston). We are nterested n the noton of congruence of such matrces. We are partcularly nterested n the adjacency matrces of semsymmetrc graphs; whch have the property (whch we wll also denote by semsymmetry) that f the th dagonal element s 0, then the th row and column are the same. Defnton 8 We say that two n n matrces A and B are congruent (denoted A = B) f there s an nvertble matrx P such that A = PBP T. One of our man results s that the congruence class of the adjacency matrx (over Z 2 ) satsfes the 2-term relaton, and so nduces a regular graph weght system. Theorem 3 For any graph G n E, adj(g) = adj( G AB ). Proof: Reorderng the vertces of G changes adj(g) by a congruence; n ths case, P s the result of dong a correspondng reorderng of the rows of the dentty matrx. So we can assume that the frst two rows and columns of adj(g) and adj( G AB ) correspond to the vertces A and B. The matrces are dentcal except for the frst row and column. Say that A has label {c, a} and B has label {c, b}, so n G AB A has label {a, b}. An entry adj( G AB ) 1 (where 1, 2) s equal to adj(g) 1 +adj(g) 2 (mod 2), and smlarly for the frst column. adj( G AB ) 12 = adj(g) 12 f and only f b = c,.e. f adj(g) 22 = 0, so adj( G AB ) 12 = adj(g) 12 + adj(g) 22. Fnally, adj( G AB ) 11 = 0 f a = b, and 1 otherwse. We have fve cases. Case 1: a = b = c. In ths case adj(g) 11 = adj(g) 22 = 0 and adj(g) 12 + adj(g) 21 = 4 (due to the semsymmetry of the graph). So, mod 2, adj( G AB ) 11 = 0 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. Case 2: a = b c. In ths case adj(g) 11 = adj(g) 22 = 1 and adj(g) 12 + adj(g) 21 = 4. So, mod 2, adj( G AB ) 11 = 0 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. Case 3: a b = c. In ths case, adj(g) 11 = 1 and adj(g) 22 = 0, and adj(g) 12 + adj(g) 21 = 2. So, 11

13 mod 2, adj( G AB ) 11 = 1 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. Case 4: b a = c. In ths case, adj(g) 11 = 0 and adj(g) 22 = 1, and adj(g) 12 + adj(g) 21 = 2. So, mod 2, adj( G AB ) 11 = 1 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. Case 5: a, b, c all dfferent. In ths case adj(g) 11 = adj(g) 22 = 1 and adj(g) 12 + adj(g) 21 = 1. So, mod 2, adj( G AB ) 11 = 1 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. We conclude that adj( G AB ) 11 = adj(g) 11 + adj(g) 22 + adj(g) 12 + adj(g) 21. Therefore adj(g) and adj( G AB ) are congruent by a matrx P, where P s the elementary matrx constructed from the dentty matrx by addng the second row to the frst row. In other words, P s the same as the dentty, except for the 2 by 2 matrx n the upper left corner, whch s [ Corollary 1 The congruence class of the adjacency matrx of the ntersecton graph s nvarant under the 4-term relatons. Corollary 2 The determnant (mod 2) and rank (over Z 2 ) of the adjacency matrx of the ntersecton graph are regular graph weght systems. Proof: If P s nvertble over Z 2, the rank of PAP T s the same as the rank of A. In addton, det(p) = det(p T ) = 1 (mod 2), so the determnant s also a congruence nvarant mod 2. ]. 4.3 Relatons to the Conway and Homfly polynomals For knots, the Conway and Homfly weght systems can be nterpreted va ntersecton graphs [13]. To what extent can we do the same for strng lnks? In general, ths s stll an open problem, but we can offer a few ntal results. Let us recall the defntons of the Conway polynomal and weght system. The Conway polynomal of a lnk s a power seres (L) = n 0 a n(l)z n. It can be computed va the sken relaton (where L +, L, L 0 are as n Fgure 12): (L + ) (L ) = z (L 0 ) { 1 f k = 1 (unlnk of k components) = 0 f k > 1 We defne the Conway polynomals of a strng lnk as the Conway polynomal of the lnk formed by closng the strng lnk n the natural way. The coeffcent a n s a fnte type nvarant of type n [1, 3], and therefore defnes a weght system b n of degree n. The collecton of all these weght systems s called the Conway weght system, denoted C. Consder a chord dagram D, together wth a chord v. Let D v be the result of surgery on v,.e. replacng v by an band whch preserves the orentaton of the components, and then removng the nteror of the band and the ntervals where t s attached to D, as shown n Fgure 13 (so D v may have Fgure 12: Dagrams of the sken relaton. 12

14 Fgure 13: Surgery on a chord v multple boundary crcles). The sken relatons for the Conway polynomal gve rse to the followng relatons for C: C(D) = C(D v ) { 1 f k = 1 C(unlnk of k components) = 0 f k > 1 It s easy to show [2] that ths weght system satsfes the 2-term relatons of secton 4. Smply surger the two chords; the 2-term relaton then says just that one band can be sld over the other, whch doesn t change the topology of the dagram. Bar-Natan and Garoufaldes also showed that the weght system for the Conway polynomal for knots s just the determnant of the adjacency matrx for the ntersecton graph, mod 2. The next theorem extends ths result to strng lnks wth two components. Theorem 4 For any chord dagram D on a strng lnk wth two components, C(D) = det(γ(d))rank(γ(d)) (mod 2). Proof: Snce both of these weght systems satsfy the 2-term relatons, t suffces to show that they agree on dagrams n normal form (see secton 4.1). For strng lnks wth two components, normal form conssts of a caravan on the frst component, and 0, 1, or 2 parallel chords between the two components. Say that the caravan has m 1 1-humped camels and m 2 2-humped [ camels, ] and there are l chords between the components (l = 0, 1, 2). Then adj(γ(d)) m2 0 1 = [1] l [0] m1. So det(γ(d)) = l+m2 0 m1 (mod 2) and rank(γ(d)) = l + 2m 2 l (mod 2). The product of the rank and determnant s 1 (mod 2) when l = 1 and m 1 = 0, and 0 otherwse. On the other hand, f we surger all the chords of the dagram, and close the strng lnk, we obtan an unlnk wth m components when l = 0, 2, and m when l = 1, whch means that C(D) = 1 only when m 1 = 0 and l = 1. So the two weght systems agree. Wth more than two components, the stuaton s more complcated; n fact, the Conway weght system s not always determned by the adjacency matrx of the ntersecton graph. Fgure 14 shows two chord dagrams D 1 and D 2 on 4 components whch llustrate ths. It s easy to see that these ntersecton graphs D 1 = D 2 = Fgure 14: C(D) s not determned by the adjacency matrx for these dagrams dffer only n the labelng of the vertces, so: 13

15 adj(γ(d 1 )) = adj(γ(d 2 )) = However, surgerng the chords n D 1 yelds a strng lnk whose closure has one component, and surgerng the chords n D 2 yelds a strng lnk whose closure has 3 components. Therefore, C(D 1 ) = 1, but C(D 2 ) = 0. However, we can at least gve a suffcent condton for C(D) to be trval whch depends only on the adjacency matrx. Proposton 1 Let D be a strng lnk chord dagram of degree k on n components. If ether of the followng condtons holds, then C(D) = 0. det(γ(d)) = 0. k + n s even. Proof: The frst condton means that, n the normal form, there wll be an solated chord n the caravan, so C(D) wll be trval. The second condton comes from observng that every surgery of a chord n D the number of components n the closure of D by 1 (ether addng or removng a component). Snce we begn wth n components, the number of components after all the chords are surgered s congruent to k+n modulo 2. So C(D) can be non-trval only f k + n 1 mod 2. Hence, f k + n s even, then C(D) = 0. Now we wll consder the (framed) Homfly weght system for strng lnks. The Homfly nvarant s the Laurent polynomal P(l, m) Z[l ±1, m ±1 ] defned by the followng sken relatons [11] (L + s the result of addng a postve knk to the lnk L): P(L + ) P(L ) = mp(l 0 ) P(L + ) = lp(l) P(L O) = l l 1 m P(L) P(O) = 1 If we make the substtutons m = e ax/2 e ax/2 and l = e abx/2, and expand the resultng power seres, we transform the Homfly polynomal nto a power seres n x, whose coeffcents are fnte type nvarants (of regular sotopy). These nvarants gve rse to regular weght systems whch we can collect together as the Homfly regular weght system H. The sken relatons above gve rse to the followng relatons for H, by lookng at the frst terms of the power seres (as before, D v s the result of surgerng the chord v n D): H(D) = ah(d v ) H(D O) = bh(d) H(O) = 1 So f D s an unlnk of k components, H(D) = b k 1. Snce the frst of these relatons s almost the same as for the Conway weght system C, the same argument shows that H satsfes the 2-term relatons. We can now consder strng lnk dagrams wth two components (as wth the Conway weght system, the Homfly weght system s not necessarly determned by the adjacency matrx of the ntersecton graph for chord dagrams on more than two components). Theorem 5 For any chord dagram D of degree k on a strng lnk wth 2 components, let r = rank(γ(d)). If r s odd, then H(D) = a k b k r ; f r s even, then H(D) = a k b k r+1. 14

16 Proof: As wth Theorem 4, t suffces to show that the weght systems agree on dagrams n normal form. Let D be the dagram wth m 1 one-humped camels and m 2 two-humped camels on component 1, and l chords between the two components (l = 0, 1, 2) (so the degree of D s k = l + m 1 + 2m 2 ). As before, adj(γ(d)) = [1] l [0] m1 [ ] m2, so the rank s r = l + 2m 2. r s even when l = 0, 2 and odd when l = 1. On the other hand, f we surger all the chords (each tme multplyng H by a), the resultng lnk has m components when l = 0, 2 (and r s even), and m components when l = 1 (and r s odd). So when r s even, H(D) = a k b m1+1 = a k b k r+1, and when r s odd, H(D) = a k b m1 = a k b k r. Remark: If we let a = 1 and b = 0, then H(D) = C(D). Accordng to Theorem 5, f D s a strng lnk chord dagram wth two components, we wll have H(D) = 0 n ths case unless r = k s odd (n whch case H(D) = 0 0 = 1). But ths s exactly the case when det(γ(d)) = 1 (snce the matrx has full rank) and rank(γ(d)) 1 mod 2, so C(D) = 1. So the formulas for the Conway and Homfly weght systems agree n ths case. 5 Further Questons and Problems There are several obvous questons and drectons for further research. Queston 1 To what extent are the Conway and Homfly weght systems for strng lnks wth more than two components determned by the ntersecton graph? Fgure 14 gves an example of chord dagrams on 4 components where the adjacency matrx of the ntersecton graph s nsuffcent to determne the Conway and Homfly weght systems. However, ths example seems to depend strongly on there beng 4 components - t s not clear how to fnd connected examples wth, say, 5 components where the adjacency matrx s nsuffcent. And certanly the adjacency matrx (as shown by the example n Fgure 14 does not contan all the nformaton of the ntersecton graph - n partcular, the nformaton contaned n the vertex labels. Ths leads to our next queston: Queston 2 Can we fnd nvarants of ntersecton graphs other than the adjacency matrx whch gve rse to weght systems? Queston 3 Can we fnd other nvarants of strng lnks whose weght systems can be computed va ntersecton graphs? The author has looked at Mlnor s homotopy nvarants for strng lnks [16], but there are many others. Queston 4 To what extent does the ntersecton graph determne the chord dagram? Ths queston s stll open for knots, as well, though some progress has been made [6, 12]. Some work has been done by the author [14] for the case when the ntersecton graph s a tree, but ths s only a bare begnnng. Queston 5 Can we generalze these constructons from strng lnks to lnks? We should be able to defne ntersecton graphs for lnks as a quotent space of the ntersecton graphs for strng lnks. It s yet to be seen whether ths s useful. 15

17 References [1] Bar-Natan, D.: On the Vasslev Knot Invarants, Topology 34, No. 2, pp , 1995 [2] Bar-Natan, D. and Garoufalds, S.: On the Melvn-Morton-Rozansky Conjecture, Invent. Math. 125, pp , 1996 [3] Brman, J. and Ln, X.S.: Knot polynomals and Vasslev s Invarants, Invent. Math. 111, pp , 1993 [4] Bouchet, A.: Crcle Graph Obstructons, Journal of Combnatoral Theory, Seres B, 60, pp , 1994 [5] Chmutov, S.V. and Duzhn, S.V.: The Kontsevch Integral, Acta Applcandae Math., Vol. 66, pp , 2001 [6] Chmutov, S.V., Duzhn, S.V. and Lando, S.K.: Vasslev Knot Invarants I, II, III, Advances n Sovet Mathematcs 21, pp , 1994 [7] Habegger, N. and Ln, X.S.: The Classfcaton of Lnks up to Lnk-Homotopy, J. of the Amer. Math. Soc., Vol. 3, No. 2, 1990, pp [8] Kontsevch, M.: Vasslev s knot nvarants, Advances n Sovet Mathematcs, Vol. 16, Part 2, pp , 1993 [9] Lando, S.: On a Hopf algebra n graph theory, Journal of Combnatoral Theory, Seres B, v.80, p , 2000 [10] Le, Thang: unpublshed, 1996 [11] Lckorsh, W.B.: An Introducton to Knot Theory, GTM vol. 175, Sprnger, New York, 1997 [12] Mellor, B.: The Intersecton Graph Conjecture for Loop Dagrams, Journal of Knot Theory and ts Ramfcatons, Vol. 9, No. 2, Also see arxv:math.gt/ [13] Mellor, B.: A few weght systems arsng from ntersecton graphs, Mchgan Math. J., Vol. 51, No. 3, pp , Also see arxv:math.gt/ [14] Mellor, B.: Tree dagrams for strng lnks, preprnt, May 2004, arxv:math.gt/ [15] Mellor, B.: Tree dagrams for strng lnks II: Determnng Chord Dagrams, preprnt, August 2004, arxv:math.gt/ [16] Mellor, B.: Weght systems for Mlnor nvarants, n preparaton [17] Meng, G.: Bracket models for weght systems and the unversal Vasslev nvarants, Topology and ts Applcatons 76, pp , 1997 [18] Morton, H.R. and Cromwell, P.R.: Dstngushng Mutants by Knot Polynomals, Journal of Knot Theory and ts Ramfcatons 5, No. 2, pp , 1996 [19] Mlnor, J. and Husemoller, D.: Symmetrc Blnear Forms, Sprnger-Verlag, New York, 1973 [20] Stanford, T.: Fnte-Type Invarants of Knots, Lnks, and Graphs, Topology 35, No. 4, pp , 1996 [21] Vasslev, V.A.: Cohomology of knot spaces, n Theory of Sngulartes and Its Applcatons (ed. V.I. Arnold), Advances n Sovet Mathematcs, Vol. 1, pp ,