MU TRANSPOSOSOME ISABEL DARCY, JOHN LUECKE, AND MARIEL VAZQUEZ

Transcription

1 MU TRANSPOSOSOME ISABEL DARCY, JOHN LUECKE, AND MARIEL VAZQUEZ Abstract. xxxx In [PJH], Pathania et al determined the shape of DNA bound within the Mu transposase protein complex using an experimental technique called difference topology [GBJ, KBS, PJH] and by making certain assumptions regarding the DNA shape. We show that their most restrictive assumption (described near the end of section 1) is not needed and in doing so, determine that the only biologically reasonable solution for the shape of DNA bound by Mu transposase is that given in [PJH] (Fig. 0.1). We will call this tangle the PJH solution. The Mu-DNA complex modeled by this tangle is called the Mu transpososome. Figure 0.1 In section 1 we provide some biology background and describe eight experiments from [PJH]. In section 2, we translate the biology problem of determining the shape of DNA bound by Mu into a mathematical model. The mathematical model consists of a system of ten 3-string tangle equations (figure 2.2). Using 2-string tangle analysis, we simplify this to a system of four tangle equations (figure 2.14). In section 3 we exhibit infinitely many different 3-string tangle solutions to these equations and characterize them in terms of knotted graphs. Except for the PJH solution, these solutions are too complicated to be biologically reasonable. However the existence of solutions different from the PJH solution raises the possibility of alternate acceptable models. We show that all solutions to the mathematical problem other than the PJH solution are too complex, where the 1

2 complexity is measured either by the rationality or by the minimal crossing number of the 3-string tangle solution. In section 3, we show that the only rational solution tangle is the PJH solution. In particular we prove Corollary 3.8. Let T be a solution tangle. If T is rational or split, or if T has parallel strands then T is the PJH solution. In section 4 we show that if T is not the PJH solution, then it must have at least 8 crossings up to free isotopy (i.e. allowing the ends of the tangle to move under the isotopy). Fixing the framing of a solution tangle (the normal framing of section 2), and working in the category of tangle equivalence i.e. isotopy fixed on the boundary we prove the following lower bound on the crossing number of exotic solutions: Proposition 5.1. Let T be a solution tangle. If T has a projection with fewer than 10 crossings, then T is the PJH tangle. Converting this result to the framing used in [PJH], if Mu binds fewer than 9 crossings, then the PJH solution is the only solution. The PJH solution has 5 crossings. We think of Corollary 3.8 and Proposition 5.1 as saying that the PJH solution is the only biologically reasonable model for the Mu transpososome. In difference topology experiments, a protein of known mechanism is used to knot or link DNA bound by the protein under study. In [PJH], (2,4) torus links were products in three of the experiments while the (2, 2) torus link resulted in a fourth experiment. We will describe 8 experiments from [PJH]. However only four of the experiments are needed for our analysis, while the remaining four experiments were used in the model design [PJH]. Our analysis focuses on the three (2,4) torus link products as well as the (2, 2) torus link product. The results in sections 2-4 hold equally well when the experimental products are (2, L) torus links [HS] or the trefoil knot [KMOS]. 1. Biology Background and Experimental Data Transposable elements, also called mobile elements, are fragments of DNA able to move along a genome by a process called transposition. Mobile elements play an important role in the shaping of a genome [DMBK, S]. They can impact the health of an organism by introducing genetic mutations. Furthermore transposition is mechanistically very similar to the way certain retroviruses, including HIV, integrate into their host genome. 2

3 Bacteriophage Mu is a system widely used in transposition studies due to the high efficiency of the Mu transposase (reviewed in [CH]). The MuA protein performs the first steps required to transpose the Mu genome from its starting location to a new DNA location. MuA binds to specific DNA sequences which we refer to as attl and attr sites (named after left and right attaching regions). A third DNA sequence called the enhancer (E) is also required to assemble the Mu transpososome. Hence the Mu transpososome consists of 3 segments of double-stranded DNA captured in a protein complex. The Mu transpososome is very stable [BM, MBM]. In this paper we are interested in studying the topological structure of the DNA within the Mu transpososome. We base our study on the experiments of [PJH] which used a technique called difference topology. In this technique, circular DNA is first incubated with the protein(s) 1 under study (in this case, MuA). These proteins bind DNA. A second protein whose mechanism is well understood is added to the reaction. This second protein is a protein that can cut DNA and change the circular DNA topology before resealing the break(s), resulting in knotted or linked DNA. DNA crossings bound by the first protein will affect the product topology. Hence one can gain information about the DNA conformation bound by the first protein by determining the knot/link type of the DNA knots/links produced by the second protein. In [PJH], Cre recombinase was used to trap crossings bound within the Mu transpososome tangle. Cre recombinase binds to specific DNA sequences called loxp sites. Cre binds to two loxp sites, cuts the DNA at both these sites and exchanges the ends before resealing the breaks. Since Cre binds two DNA segments, the Cre mechanism can be modeled using 2-string tangles. A possible 2-string tangle model for the local Cre mechanism is shown in fig. 1.1 [GGD]. Proper DNA sequence orientation within the Cre-DNA complex is a requirement for Cre recombination [LTJ, HL]. 1 Although we use the singular form of protein instead of the plural form, most protein-dna complexes involve several proteins. For example formation of the Mu transpososome involves four MuA proteins and the protein HU. Also since these are test tube reactions and not single molecule experiments, many copies of the protein are added to many copies of the DNA substrate to form many complexes. 3

4 Figure 1.1 In the difference topology experiments described in [PJH], circular unknotted DNA is created containing the three binding sites for the Mu transposome (attl, attr, E) and two binding sites for Cre (two loxp sites). We will refer to this unknotted DNA as substrate. The circular DNA is first incubated with the proteins required for Mu transposition, thus forming the transpososome complex. This leaves three DNA loops free outside the transpososome. The two loxp sites were strategically placed in two of the three outside loops. The complex is incubated with Cre enzymes, which bind the loxp sites, introduce two double-stranded breaks, recombine the loose ends and reseal them. This cut-andpaste reaction may change the topology (knot/link type) of the DNA circle. Changes in the substrate s topology resulting from Cre action can reveal the structure within the Mu transpososome. By looking at such topological changes, Pathania et al. [PJH] deduced the structure of the transpososome to be the PJH solution (Figure 0.1). In this paper we give a knot theoretic analysis that supports this deduction. We show that although there are other configurations that would lead to the same product topologies seen in the experiments, they are necessarily too complicated to be biologically reasonable. Tangle analysis is a mathematical method that models an enzymatic reaction as a system of tangle equations [ES1, SECS]. 2-string tangle analysis has been successfully used to solve the topological mechanism of several site-specific recombination enzymes [ES1, ES2, SECS, GBJ, D, VS, VCS, BV]. The Mu transpososome is better explained in terms of 3-string tangles. Some efforts to classifying rational 3-string tangles and solving 3-string tangle equations are underway [C1, C2, EE, D1]. In this paper we find tangle solutions for the relevant 3-string tangle equations. We show that the PJH solution (Figure 0.1) is the unique rational solution, and we characterize solutions in terms of certain knotted graphs called solution graphs. The unknotted substrate captured by the transpososome is modeled as the union of the two 3-string tangles T 0 T where T is the transpososome tangle and T 0 is the tangle outside the transpososome complex. T 0 T is represented in Fig Notice that in 4

5 this figure the loxp sites are placed on either side of the enhancer sequence, but the placement of these sites varies throughout the experiments. lox p enhancer lox p attl attr Figure 1.2 If the orientation of both loxp sites induces the same orientation on the circular substrate (in biological terms, the sites are directly repeated), then recombination by Cre results in a link of two components and is referred to as a deletion (figure 1.3). Otherwise the sites are inversely repeated, the product is a knot, and the recombination is called an inversion (figure 1.3). = = Direct Repeats Inverted Repeats Figure 1.3 In [PJH] six out of eight experiments were designed by varying the relative positions of the loxp sites and their relative orientations. The last pair of experiments involved deleting one of the Mu binding sites and placing that site on a linear piece of DNA to be provided in trans as described shortly. In the first pair of experiments from [PJH], loxp sites were introduced in the substrate on either side of the enhancer sequence (E) (fig. 1.2). The sites were inserted with orientations to give, in separate runs, deletion and inversion products. The transpososome was disassembled and the knotted or linked products analyzed using gel electrophoresis and electron microscopy. The primary inversion products were (+) trefoils, and the primary deletion products were 4-crossing right-hand torus links ((2, 4) torus links). 5

6 The assay was repeated, but now the loxp sites were placed on either side of the attl sequence. The primary products were (2, 4) torus links for deletion, and trefoils for inversion. In a third set of experiments, the assay was repeated again with the loxp sites on either side of the attr sequence. The primary products were (2, 4) torus links for deletion, and 5-crossing knots for inversion. A final pair of experiments were done, the in trans assay. Circular DNA substrates were created that contained attl and attr sites but no enhancer site. Each loxp site was inserted between the attl and attr sites as shown in Figure 1.4. The circular substrate was incubated in solution with linear DNA molecules containing the enhancer sequence and with the proteins required for transpososome assembly. The loxp sites in the resulting transpososome complex underwent Cre recombination. After the action of Cre and the disassembly of the transpososome (including the removal of the loose enhancer strand), the primary inversion products were trefoil knots, and the primary deletion products were (2,2) torus links (Hopf links). loxp enhancer E attl loxp attr Figure 1.4 The results of these experiments are summarized in Table Vertical columns correspond to the placement of loxp sites, e.g. the attl column shows inversion and deletion products when the loxp sites were placed on either side of the attl sequence. 2 The chirality of the products was only determined when the loxp sites were placed on either side of the enhancer sequence. We here assume the chirality in Table 1.5, where (2,4) torus link denotes the 4-crossing right-hand torus link. If any of the products are left-handed (2, 4) torus links, the results of sections 2-5 applied to these products leads to biologically unlikely solutions. 6

7 enhancer attl attr in trans Inversion (+)-trefoil trefoil 5-crossing knot trefoil Deletion (2, 4)-torus (2, 4)-torus (2, 4)-torus (2, 2)-torus link link link link Table 1.5 Figure 1.6 shows the action of Cre on the transpososome proposed in [PJH] to produce the observed products. For example, E-inversion refers to the product corresponding to loxp sites introduced on either side of the enhancer sequence, where the loxp are inversely repeated. However, there are other 3-string tangles, assuming the same action of Cre, that give rise to the same products. Figure 1.7 shows one such example. If one replaces the tangle of Figure 0.1 with that of Figure 1.7 in Figure 1.6, the captions remain valid. 7

8 L E R E-inversion = trefoil E-deletion = (2,4) torus L-inversion = trefoil L-deletion = (2,4) torus R-inversion = (2,5) torus R-deletion = (2,4) torus in trans -inversion = trefoil in trans -deletion = Hopf link Figure 1.6 Figure 1.7 [PJH] determined the shape of DNA within the Mu transpososome to be the PJH solution (Figure 0.1) by making a rather restrictive assumption regarding this DNA conformation. They looked at only the most biologically likely shape: a 3-branched 8

9 supercoiled structure like that shown in Figure?? 3. The loxp sites were strategically 4 placed close to the Mu transposome binding sites in order to prevent Cre from trapping random crossings not bound within the Mu transposome. In half the experiments it was assumed that Cre trapped one extra crossing outside of the Mu transpososome in order to obtain proper loxp sequence orientation (Figure 1.1), and it was also assumed that this occurred with the higher crossing product when comparing inversion versus deletion products. Hence a crossing outside of the Mu transposome can be seen in Figure 1.6 in the case of E-deletion, L-deletion, R-inversion, and in trans-inversion. In all other cases, it was assumed that Cre did not trap any extra crossings outside of the Mu transpososome. By assuming a branched supercoiled structure, [PJH] used their experimental results to determine the number of crossings trapped by Mu in each of the three branches. In our analysis, we do not assume a branched supercoiled structure within the Mu transpososome. We show in the following sections that we are able to reach the same conclusion as [PJH] without this assumption. Will add supercoiled figure. 2. Normal Form 2.1. Normal form. The substrate for Cre recombination in [PJH] is modeled as the union of two 3-string tangles T T 0, where T represents the transpososome tangle, and T 0 is the tangle outside the transpososome. We here introduce a framing for T called the normal form which is different from that in the PJH solution (section 1). The choice of framing affects only the arithmetic in section 2. as discussed at the end of this section. The choice of framing does not affect any of the results in sections 3 or 4. The results of section 5 on the crossing number of T are made with respect to this framing. First we assume there is a projection of T T 0 as in Figure 2.1. Let c 1, c 2, c 3 be the strings of T 0 and s 12, s 23, s 31 be the strings of T. The substrate is the union of the c i s and the s ij s. Take a projection of this union so that c 1, c 2, c 3 are isotopic (relative endpoints) onto the tangle circle and so that the endpoints of s ij are contiguous on the tangle circle 3 Figure of supercoiled DNA courtesy of. This conformation was generated using a computer simulation modeling unknotted circular DNA in solution or is an AFM image. 4 Although we described only 8 experiments from [PJH], they performed a number of experiments to determine and check effect of site placement. 9

10 (Figure 2.1). Note that this is a different projection than the projection in [PJH]. It is a simple matter to convert between projections, as described below. S 23 c 2 c 3 0 S 12 S 31 c 1 Figure 2.1 In each experiment the two recombination sites for Cre (loxp sites) are located on two strings c i and c j (i j). Cre bound to a pair of strings c i c j can be modeled as a 2-string tangle P of type 0. Earlier studies of Cre support the assumption that Cre 1 recombination takes P = 0 into R = 1 (figure 1.1) [PJH, GBJ, GGD, KBS]. However, it 1 0 is possible that Cre recombination traps crossings outside of the Mu transposome. Hence the system of tangle equations shown in fig 2.2 can be used to model these experiments where d i, v i, d t, v t represent non-trivial topology trapped via Cre recombination, but not bound by Mu. Tangle equation (1) corresponds to the unknotted substrate equation from the first six experiments. Equations (2)-(4) correspond to three product equations modeling the first three deletion experiments, while equations (5)-(7) correspond to the three product equations modeling the first three inversion experiments. Equation (8) corresponds to the unknotted substrate equation for the two in trans experiments while equations (9) and (10) correspond to the product equations modeling in trans deletion and in trans inversion, respectively. 10

11 1 d 3 = right handed (2, 4) torus link [2] 1 d 2 = right handed (2, 4) torus link [3] [1] = unknot 1 v 3 1/d 1 = right handed (2, 4) torus link [4] = trefoil [5] 1 v 2 = 5 crossing knot [6] 1/v 1 = trefoil [7] [8] In trans: = unknot 1 d t = (2, 2) torus link [9] 1 v t =trefoil [10] 1 where n = if n > 0 or if n < 0 Figure

12 Choosing a framing for the tangle T, beyond Figure 2.1, is equivalent to choosing specific values for d 1, d 2, d 3. For biological reasons, [PJH] chose d 1 = 1, d 2 = 0, d 3 = 1. For mathematical convenience, we choose d 1 = d 2 = d 3 = 0. We define the normal form equations to be the system of equations in fig 2.2 where d 1 = d 2 = d 3 = 0 and a normal form solution tangle to be a solution to the normal form equations. Note that in the normal form equations, the action of Cre deletion results in replacing P = c i c j = 0 1 with R = 1 0 If we wish to instead assume the P = c i c j = 0/1 is replaced by R = 1 d k for arbitrary d k, we can easily convert between solutions. Suppose T is a solution to this non-normal form system of equations. We can add n i twists to T at c i, for each i where n i + n j = d k by pushing n i twists of R inside T (figure 2.3). Hence T with n i crossings added inside T at c i for each i is a solution to the normal form equations. Note the n i are unique. Hence similarly, if T is a solution to the normal form equations, we can add n i twists to T at c i to obtain a solution to the non-normal form equations for arbitrary d k. Remark. The convention of [PJH] is that d 1 = 1, d 2 = 0, d 3 = 1. Hence n 1 = 0, n 2 = 1, n 3 = 0. This corresponds to adding a right-handed twist at c 2 to a solution in the PJH convention (such as Figure 0.1) to obtain a normal form solution (such as Figure 2.4). Conversely, by adding a single left-handed twist at c 2 to a normal form solution, we get the corresponding solution in the PJH convention (e.g. from the normal form solution of Figure 2.4 to the PJH solution of Figure 0.1). In normal form, the assumptions for the first three Cre deletion experiments can be summarized as: (2.1) Cre acts on the pair of strings c i, c j in the outside ball such that (1) The action of Cre on the tangle c i c j is disjoint from the isotopy of c k to the tangle circle; (2) The action of Cre on c i c j results in replacing the 0 1 tangle with

13 x 12 n 3 n 2 n 1 Figure 2.3 Remark. We will show that knowing the number of crossings trapped by Cre outside of the Mu transpososome in the first three deletion experiments is sufficient for determining the number of crossings trapped outside the Mu transpososome in all experiments. In other words choosing d 1, d 2, d 3 determines v 1, v 2, v 3, v t, and d t. Next we apply traditional 2-string tangle calculus [ES1, S] to the equations arising from the Cre deletion (not in trans) experiments First three Cre deletion tangle equations. With abuse of c i notation let the tangle circle be the union of arcs c 1 x 12 c 2 x 23 c 3 x 31 (Figure 2.4). We think of these arcs as providing a framing for the tangle T. x 23 c 2 c 3 x 31 c 1 Figure 2.4 Let O i = T c i be the 2-string tangle obtained by capping off s ji s ik with c i. Then s kj is one of the strings of the 2-string tangle O i, let ŝ i denote the other (Figure 2.5) 13

14 s 23 O 1 = c 1 s 1 Figure 2.5 By capping T along each c i we approach the problem of finding all possible 3-string tangles for the Mu transpososome by first solving three different systems of two 2-string tangle equations (one for each i). Figure 2.6 illustrates the definitions of 2-string tangle addition (+) and numerator closure (N). It also shows one of the systems of two 2-string tangle equations. T c 1 is represented by the blue and green 2-string tangle while the smaller pink 2-string tangle represents Cre. A C A = = A + C N(A) N( + 0/1) = N(1/0) -> N( + 1/0) = N(4/1) Figure 2.6 Lemma 2.1. (Cre deletion, not in trans) Let T be a normal form solution tangle and consider the three Cre deletion experiments which convert unknotted substrates to righthand (2, 4) torus links. Let O i = T c i be the 2-string tangle obtained by capping T along c i. Then O i is the 1/4 tangle, i = 1, 2, 3. Proof. All the Cre deletion events, modeled by replacing P = c j c k = 0 1 by R = 1 0 in normal form (see Figure 2.7), lead to identical systems of two 2-string tangle equations: N(O i ) = unknot N(O i + 1 ) = (2, 4) torus link 0 By [HS], O i is a rational tangle. By tangle calculus [ES1], the system admits a unique solution O i = 1/4, i = 1, 2, 3. 14

15 c j c i c i c k unknot torus link Cre inversion Figure 2.7 Each Cre inversion experiment is also modeled as a system of tangle equations: N(O i ) = N(O i + 1 v i ) = unknot inversion product If we assume that O i is constant for each pair of deletion/inversion experiments obtained by capping off each c i, then by lemma 2.1, O 1 = O 2 = O 3 = 1/4. Under these assumptions, v i can be determined by tangle calculus if the inversion products are known. From [PJH] we assume that L-inversion and E-inversion produce the (+)trefoil. Also R- inversion produces a 5-crossing knot (see Table 1.5 ), which must be a 5-torus knot since O 2 = 1/4. We assume this is the (+)5-torus knot for if it were the (-)5 torus knot, then v 2 = 9 which is biologically unlikely. Hence the only biologically reasonable solutions to the inversion equations are: L-inversion, P = 0 1 and R = 1, product =(+)trefoil; 1 (2.2) R-inversion, P = 0 1 and R= 1, product = (+)5-torus knot; 1 E-inversion, P = 0 1 and R = 1, product = (+)trefoil 1 Other possible solutions exist if O i is allowed to change with respect to the deletion versus inversion experiments Linking number considerations. We use the sign convention in figure 2.8 to determine linking numbers in lemma

16 ( + ) ( - ) Figure 2.8 Lemma 2.2. Let ˆx i be the arc on the tangle circle given by x ji c i x ik. Assume T is a normal form solution tangle. Fix an orientation on the tangle circle and use this to induce orientations on the relevant links. If Cre deletion on unknotted substrates produces (2, 4) torus links, then lk(x jk s jk, ŝ i ˆx i ) = 2 {i, j, k} = {1, 2, 3} lk(x ij s ij, x ki s ki ) = 1 Proof. The first equation follows from Lemma 2.1. To get the second equation we note that ŝ i ˆx i can be obtained via a banding connecting x ij s ij with x ki s ki along c i as indicated in Figure 2.9. s jk s ij s ki s ij s ki x ij x ki x ij x ik c i c i banding Figure 2.9 Then for {i, j, k} = {1, 2, 3}, 2 = lk(x jk s jk, ŝ i ˆx i ) = lk(x jk s jk, x ij s ij ) + lk(x jk s jk, x ki s ki ). Solving three equations ({i, j, k} = {1, 2, 3}), we obtain lk(x ij s ij, x jk s jk ) = 1. We will now see how the Cre deletion results extend to the analysis of the remaining experiments (in trans deletion and inversion) In trans deletion experiments. In the in trans portion of [PJH] (described in section 1) the enhancer sequence is not incorporated into the circular DNA substrate but remains separate in solution on its own linear molecule when the transpososome is formed 16

17 (Figure 2.10). Assuming the transpososome complex forms the same 3-string tangle in this context, the transpososome writes the substrate as a union of two 2-string tangles: T 0, a trivial tangle outside the transpososome; and T = T s 23, the tangle formed by the attr and attl strands. c 1 Figure 2.10 Lemma 2.3. (in trans Cre deletion experiment) Let T be a normal form solution tangle and let d t be as in Figure 2.2. Let T = T s 23 where s 23 is the enhancer strand in T. Then d t = 0, 4 and T s 23 = 1 2. Proof. When Cre acts on the loxp sites in the in trans experiment we assume that it takes the 0 tangle to 1 1 d t as illustrated in Figure The Cre deletion product in trans is a Hopf link (i.e. a (2, 2) torus link). Hence we are solving the 2-string tangle equations, N((T s 23 )+ 1 0 ) = unknot, N((T s 23)+ 1 d t ) = (2, 2) torus link. By [BL] T s 23 is rational, and by tangle calculus [ES1],T s 23 = 1 ±2 d t. Hence lk(x 31 s 31, x 12 s 12 ) = ±2 dt. 2 By lemma 2.2, lk(x 31 s 31, x 12 s 12 ) = 1. Hence ±2 dt 2 = 1. Thus d t = 0, 4 and T s 23 = s - s = unknot Hopf link Figure

18 In trans Cre inversion: If T s 23 = 1, then the in trans Cre inversion product is 2 N(T s v t ) = N( v t ) = (2, 2 v t ) torus knot. A (2, 3) torus knot in trans inversion product implies v t = Summary. The next proposition summarizes the results in this section. Proposition 2.4. Let T be a normal form solution tangle and s 23 correspond to the enhancer strand in the [PJH] experiments. x 23 c c 3 2 s 23 x 12 s 12 s 31 x 31 Then in the Cre deletion experiments c 1 Figure 2.12 (2.3) O i = T c i = 1 4, T s 23 = 1 2 The deletion reaction of Cre in trans (outside of T ) is modeled by replacing c 1 (c 2 x 23 c 3 ) = 0 1 by the 1 0 or 1 4 tangle (i.e. d t = 0, 4, see Figure 2.13). Finally, any 3-string tangle as in Figure 2.12 satisfying (2.3) is a normal form solution tangle. Figure 2.13 Proposition 2.4 can be generalized to products of deletion that are (2, L i ) torus links, and a (2, L t ) torus link for the in trans experiment: Proposition 2.5. Let T be in normal form and s 23 correspond to its enhancer strand. If the first three Cre deletion experiments (not in trans) produces (2, L i ) torus links, L i ±2, for i = 1, 2, 3, and if Cre deletion in trans results in (2, L t ) torus link, then (2.4) O i = T c i = 1 1, T s 23 =. L i 18 L 2 +L 3 L 1 2

19 If L t = 2, then d t = L j+l i +L k ±4 2. If L t 2, then d t = L j+l i +L k 2L t 2. Note similar results hold if L i = ±2 for i = 1, 2, and/or 3, but as this breaks into cases, we leave this for the reader. Definition. A 3-string tangle T framed as in Figure 2.12 is called a solution tangle iff it satisfies the condition (2.3) of Proposition 2.4. See Figure = = = = Figure 2.14 Proposition 2.4 reduces the study of synapse tangles to that of solution tangles. In section 3 we classify solution tangles in terms of solution graphs and use this to show that the only solution tangle that is rational is the PJH tangle. Remark. Sections 3 and 4 rely on the fact the tangles in Figure 2.14 are all of the form 1 m i. This would still be the case no matter the choice of framing or the type of (2, L i ) torus link products. Hence the results of sections 3 and 4 apply more generally. Remark. The proof of Propositions 2.4 and 2.5 use the deletion products only. As the remaining analysis depends only on Proposition 2.4, we see that the inversion products are unnecessary for our analysis. However, the inversion products were used to determine the framing in [PJH]. 19

20 3. Solution graphs Definition. A 3-string tangle is standard iff it can be isotoped rel endpoints to Figure 3.1. A standard tangle with n i = 2 is called the PJH tangle. n 2 n 3 n i n 1 = = n i right-handed twists Figure 3.1 Lemma 3.1. If a solution tangle is standard then it is is the PJH tangle. Proof. Write and solve the three equations, n i + n j = 4, describing the integral tangles resulting from capping along individual c k. Remark. The PJH tangle corresponds to the transpososome configuration deduced in [PJH], under our (normal) framing convention. Definition. The abstract wagon wheel is the graph of Figure 3.2. The vertices are labelled v 1, v 2, v 3 ; the edges labelled e 1, e 2, e 3, b 12, b 23, b 31. A wagon wheel graph, G, is a proper embedding of the abstract wagon wheel into B 3. See Figure 3.3. Two wagon wheel graphs are the same if there is an isotopy of B 3 which is fixed on B 3 taking one graph to the other. Definition. If a properly embedded graph lies in a properly embedded disk in the 3-ball we call it planar. Figure 3.4 gives examples of planar and non-planar wagon wheels. Note that the non-planar graph contains the knotted arc e 2 b 23 e 3. b 23 e 2 v e 2 v 3 3 e 2 v e 3 3 b 12 v 1 e 1 b 31 e 1 v 2 v 1 b 12 Figure 3.2 Figure

21 e e e b 23 e3 b 12 e 1 e 1 planar non-planar Figure 3.4 Definition. Let G be a wagon wheel graph. Then N(G) denotes a regular neighborhood of G in the 3-ball. Let J 1, J 2, J 3 be meridian disks of N(G) corresponding to edges e 1, e 2, e 3. Let J i = J i. Let Γ 12, Γ 23, Γ 31 be meridian disks of N(G) corresponding to edges b 12, b 23, b 31. Let γ ij = Γ ij. See Figure 3.5. Γ Γ 12 Γ 31 N(G) 1 Figure 3.5 Definition. X(G) = exterior of G = B 3 (N(G) N( B 3 )). Note that X(G) is a surface of genus 3. Definition. G is a solution graph iff it is a wagon wheel graph with the property that deleting any one of the edges {e 1, e 2, e 3, b 23 } from G gives a subgraph that is planar. The graphs G in Figure 3.6 are solution graphs. Definition. Let G be a wagon wheel graph. A 3-string tangle T is carried by G, written T (G), iff its 3-strings s 12, s 23, s 31 can be (simultaneously) isotoped to lie in N(G) such that 1) s ij intersects each of J i, J j once 2) s ij intersects γ ij once and is disjoint from the remaining γ st. 21

22 Figures 3.6 gives examples of tangles carried by wagon wheel graphs. G (G) e 2 e 3 e 1 G (G) Figure 3.6 Lemma 3.2. If T 1, T 2 are both carried by G, then they are isotopic (rel endpoints) up to twisting N(G) along the J i (i.e., they differ by twists at the end). Proof. Left to the reader. The following lets us work with solution graphs rather than solution tangles. solution graph economically encodes the conditions needed for a solution tangle. The Theorem 3.3. Let T be a solution tangle, then T is carried by a solution graph. Conversely, a solution graph carries a unique solution tangle. Proof. Let T be a solution tangle with strands s 12, s 23, s 31. We show that T is carried by a solution graph G. Let C = c 1 s 12 c 2 s 23 c 3 c 31, recalling that the c i are the capping arcs of Figure For each i = 1, 2, 3, let P i be a point in c i. Isotope C into the interior of B 3 by pushing each c i slightly to the interior. Under the isotopy P i traces 22

23 out an arc e i from P i B 3 to C. c 3 e 2 e 3 c 2 c 1 C Figure 3.7 e 1 G Define G = C e 1 e 2 e 3. See Figure 3.7. G is a wagon wheel graph that carries T. Note that the strings s ij of T correspond to the edges b ij of G. The conditions for T to be a solution tangle now correspond to those defining G as a solution graph. For example, G b 23 corresponds to the graph (s 12 c 1 s 13 ) e 1. As T is a solution tangle, s 12 c 1 s 13 e 1 lies in a properly embedded disk D in B 3. Hence G b 23 lies in D, and is planar. We now show that a solution graph carries a unique solution tangle. By Lemma 3.2 the tangles carried by G are parameterized by integers n 1, n 2, n 3 as in Figure 3.8. Denote these tangles T (n 1, n 2, n 3 ). As G is a solution graph T (0, 0, 0) c i will be the 2-string integral tangle f i in the framing with c j = 1/0, x jk = 0/1, where f i N. Then T (n 1, n 2, n 3 ) c i will be the 2-string tangle f i + n j + n k where {i, j, k} = {1, 2, 3}. n 2 n 3 N(G) n 1 Figure

24 We choose n i, n j, n k to satisfy n 1 + n 2 + f 3 = 4 n 2 + n 3 + f 1 = 4 n 3 + n 1 + f 2 = 4 Since f 1 + f 2 + f 3 is even, this has a unique integer solution. Then T (n 1, n 2, n 3 ) c i = +4 tangle for each i. Now G b 23 planar guarantees that the 2-string tangle T (n 1, n 2, n 3 ) s 23 = f for some integer f in the framing c 1 = 1/0, x 12 = 0/1. We have chosen n 1, n 2, n 3 so that lk(x ij s ij, ŝ i ˆx i ) = 2. The argument of Lemma 2.2 shows that lk(x 12 s 12, x 31 s 31 ) = 1. Thus T (n 1, n 2, n 3 ) s 23 = +2 tangle, and T is a solution tangle. The uniqueness of T follows from the parametrization of the tangles by the n i and the fact that T c i = +4 tangle. Theorem 3.3 allows us to construct infinitely many solution tangles, via solution graphs. Recall that a Brunnian link, L, is one such that once any component is removed the remaining components form the unlink. By piping a Brunnian link to a given solution graph, we generate a new solutions graph as pictured in Figure 3.9 (where we have begun with the second solution graph of Figure 3.6). e 2 b 23 e 3 e 1 Figure 3.9 The following lemma furthers the philosophy that solution tangles correspond to solution graphs. Lemma 3.4. If T is carried by a planar wagon wheel graph G then T is standard. Proof. There is an isotopy of B 3, keeping the boundary of B 3 fixed, taking G to the planar wagon wheel graph of figure 3.2. Apply Lemma 3.2 to see T as standard. 24

25 We now state the two main theorems and their consequences. Recall that a surface F (F is not a 2-sphere or disk) in a 3-manifold M is compressible if there is a disk D embedded in M such that D F = D and D does not bound a disk in F. F is incompressible otherwise. Theorem 3.5. Let G be a solution graph and F = X(G) ( i J i γ 23 ), in the notation set at the beginning of this section. If F is compressible in X(G), then G is planar. Theorem 3.6. Let G be a solution graph. Then either X(G) is incompressible in X(G) or G is planar. Definition. Let T be a 3-string tangle. T is rational if there is an isotopy of B 3 taking T to the standard tangle of Figure 3.10(a). T is split if there is a disk separating the strands of T (Figure 3.10(b)). Strands s 1, s 2 of T are parallel if there is a disk D in B 3 such that int(d) is disjoint from T and D = s 1 α s 2 β where α, β are arcs in B 3 (Figure 3.10(c)). s 1 s 2 standard tangle split parallel D Figure 3.10 Note: A rational tangle is split (and has parallel strands). Definition. Let T be a 3-string tangle with strings s 1, s 2, s 3. X(T ) = B 3 nbhd(s 1 s 2 s 3 B 3 ). Lemma 3.7. If T is carried by G then (X(G), X(G)) is isotopic to (X(T ), X(T )). Proof. Left to the reader. Corollary 3.8. Let T be a solution tangle. If T is rational or split or if T has parallel strands, then T is the PJH tangle. Proof. Let G be the solution graph carrying T. If T is split (e.g., rational) or if it has parallel strands then X(T ) is compressible in X(T ). By Lemma 3.7, X(G) is 25

26 6 compressible in X(G). By Theorem 3.6, G is planar. By Lemma 3.4, T is standard. Now Lemma 3.1 says T is the PJH tangle. Now we give the proofs to Theorems 3.5 and 3.6. Proof of Theorem 3.5. Assume there is a disk D, properly embedded in X(G) such that D F and D does not bound a disk in F. Write B 3 = X(G) X(G) M where M = nbhd(g B 3 ). Let M nbhd(j 1 J 2 J 3 Γ 23 ) = M 1 M 2 where M 1 is S 2 I and M 2 is a 3-ball. D lies in M i for some i, hence we can form a 2-sphere, S, by capping off D with a disk in M i (take the disk bounded by D on M i and push in slightly). That is, S is a 2-sphere which intersects X(G) in D. If D lay on M 1, then S would have to be non-separating (there would be an arc from J 1 to J 2, say, on M 1 which intersected D once and an arc connecting J 1 to J 2 on M 2 which missed D, hence S, altogether). But 2-spheres in a 3-ball are separating. D J 2 γ 23 J 3 d J 1 Figure 3.11 Thus D must lie in M 2. But then in M 2, D cannot separate J 1, J 2, or J 3, else S would be non-separating. Furthermore the two copies of Γ 23 on M 2 cannot be separated, else S is non-separating. But this means that S can be isotoped to intersect N(G) in two copies of Γ 23, Γ 23 and Γ 23, connected by a band d lying in N(G). See Figure Then S is a 2-sphere which intersects G in only two points and separates off a subarc b 23 of b 23. Now b 23 must be unknotted in the 3-ball bounded by S, otherwise deleting e 1 would not give a planar graph. Hence, there exists a disk, D in the 3-ball bounded by S such that the boundary of D consists of the arc b 23 and an arc in Γ 23 Γ 23 d. Then D can be taken to be a disk, properly embedded in X(G) whose boundary is disjoint from J 1 J 2 J 3 and whose boundary intersects γ 23 exactly once. Let G = G b 23. Let H = nbhd(b 23 ). Then N(G) is obtained by adding the 1-handle H to N(G ) where the attaching disks lie in the annular regions of N(G ) between 26

27 { J 3 γ 31 and between J 2 γ 12. Let P be the 5-punctured sphere component of X(G) nbhd( i J i ) H. Label the components of P by J 1, J 2, J 3, γ, γ. G is a solution graph so G lies in a disk properly embedded in B 3. The disk gives a framing to N(G ) nbhd(j 1 J 2 J 3 ). We use this framing in Figure J 2 γ" α γ J3 J 1 Figure 3.12 Because D intersects γ 23 exactly once we may isotop D so that D = α β where α P, β H, and β is isotopic to a core of H. Claim 3.9. After an isotopy of D in X(G), up to symmetry (switching indices 2 and r 3), we may assume that α P is either as in Figure 3.13, where, s 2n+1 2m+1 done in a neighborhood of γ 12, γ 31 for some r, s Z. twisting is r twist 2n+1 times s twist 2m+1 times J J 3 2 n m } γ 12 γ 31 J 1 Figure 3.13 Proof. Isotop α to minimize α γ 31, α γ 12. Note that essential arcs in a 3-punctured sphere which are disjoint from a single -component are isotopic to the following standard 27

28 classes (rel endpoints) up to Dehn twisting in the remaining -components: Figure 3.14 Finally, note that the Dehn twisting along γ, γ can be pushed into H. This alters β, but it remains isotopic to a core of H. Claim If m 0, then K = G e 2 b 31 is knotted. Recall that edges e i, b jk of G correspond to meridians J i, Γ jk of N(G). Let C be the 3-ball component of N(G ) ( i J i) that contains P. Then C can be pictured as a 2-sphere intersecting K in 4 pts and dividing K into two arcs e 1 b 12, e 3 on the inside and b 23 on the outside (because β is a core of H, we are using D to identify b 23 with α). b e 3 b 12 e 1 + Figure C K = 4 (marked by x s) Thus capping off K with a final arc on B 3, gives a knot ˆK in B 3 which is the numerator closure of the tangle whose slope is defined by b 23 = α on C. Note that γ 31 frames this numerator closure and that α intersects γ 31 essentially 2m + 1 times. If m 0, ˆK is a nontrivial 2-bridge knot, consequently K is a knotted arc in B 3. Since G e 2 is planar, so is G e 2 b 31 = K. Thus m = 0 in Figure 3.13(for the symmetric case, G e 3 planar implies m = 0). Again, we may push α off of N(G ) to 28

29 represent b 23. Since G b 23 is planar, we have G as v 2 v 3 v 1 Figure 3.16 After an isotopy of B 3 (rel B 3 ), i.e., twisting along v 1, v 3, G is standard. Q.E.D. (Theorem 3.5) Remark. We use that G is a solution graph in Theorem 3.5 to rule out examples like the following: b 23 e 2 b 31 e3 b 12 e 1 Figure 3.17 Here X(G) ( i J i γ 23 ) is compressible, but G is not standard (and not a solution graph) because G e 2 b 31 is knotted. Note that this graph satisfies all but one of the conditions of a solution graph (G e 2 is not planar). Before beginning the proof of Theorem 3.6 we recall that Handle Addition Lemma. Definition. Let M be a 3-manifold, F a subsurface of M, and J a simple closed curve in F. Then τ(m, J) is the 3-manifold obtained by attaching a 2-handle to M along J. That is, τ(m, J) = M J H, where H is a 2-handle. σ(f, J) is the subsurface of τ(m, J) obtained by surgering F along H. That is, σ(f, J) = (F H) int(f H). Handle Addition Lemma. ([CG], or Lemma of [CGLS]) Let M be an orientable, irreducible 3-manifold and F a surface in M. Let J be a simple, closed curve in F. 29

30 6 Assume σ(f, J) is not a 2-sphere. If F is compressible in M but F J is incompressible, then σ(f, J) is incompressible in τ(m, J). Proof of Theorem 3.6. Assume X(G) compresses in X(G). Note that X(G) is irreducible. Let F be the 5-punctured sphere component of X(G) nbhd(j 1 J 2 J 3 γ 23 ). See Figure J 2 J3 F J 1 Figure 3.18 Claim F 1 = X(G) γ 23 is compressible in X(G). Proof. Since G b 23 is planar,τ(x(g), γ 23 ) = X(G b 23 ) has compressible boundary. Hence Claim 3.11 follows from the Handle Addition Lemma. Claim F 2 = F 1 J 1 is compressible in X(G). Proof. By Claim 3.10, F 1 is compressible in X(G). On the other hand, since τ(x(g), J 1 ) = X(G e 1 ) and G e 1 is planar, σ(f 1, J 1 ) compresses in τ(x(g), J 1 ), see Figure The claim now follows from the Handle Addition Lemma. D = compressing disk D J 2 J 3 Figure 3.19 Claim F 3 = F 2 J 2 is compressible in X(G). 30

31 Proof. Since G e 2 is planar, σ(f 2, J 2 ) is compressible in τ(x(g), J 2 ) (as in Figure 3.19). Thus Claim 3.11 with the Handle Addition Lemma implies the Claim. Finally, as G e 3 is planar, σ(f 3, J 3 ) is compressible in τ(x(g), J 3 ) = X(G e 3 ). Claim 3.12 with the Handle Addition Lemma says that F 3 J 3 = X(G) (γ 23 J 1 J 2 J 3 ) is compressible in X(G). By Theorem 3.5, G is planar. Q.E.D. (Theorem 3.6) 4. Cr(T ) 8 up to free isotopy Definition. Two 3-tangles T 1, T 2 are freely isotopic if there is an isotopy of the 3-ball, which is not necessarily fixed on its boundary, taking T 1 to T 2. A rational tangle is a tangle which can be freely isotoped to one with no crossings. In the last section we showed that if T is a solution tangle which is rational, then it must be the PJH tangle. Here we generalize this by showing that if T can be freely isotoped to have fewer than seven crossings, then it must be the PJH tangle. Lemma 4.1. If one string of a 3-tangle T intersects the union of the other two strings at most once, then T is split. Proof. If the string passes over the union isotop the strand to the front hemisphere of the tangle sphere, otherwise isotop to the back. Definition. To a projection of a 3-tangle T we associate the 4-valent graph Γ(T ), that is obtained by placing a vertex at each crossing. If T is not split, we may label e 1,..., e 6 the edges which are incident to the tangle circle. Let v 1,..., v 6 be the vertices of Γ(T ) which are endpoints of e 1,..., e 6. v 1 e 1 v 3 e 3 Lemma 4.2. Assume T is not split. If v i = v j for some i j, then the crossing number of its projection can be reduced by free isotopy. Proof. Assume v i = v j with i j. If e i, e j are opposing at v i, then e i e j is a string of T intersecting the other strings exactly once. By Lemma 4.1, T is split. So we assume 31

32 e i, e j are not opposing at v i. e i v i = v j e j But then we can untwist e i, e j to reduce the crossing number. Definition. If T is not split, let f i be the face of Γ(T ) containing e i, e i+1. Lemma 4.3. Assume T is not split. No two edges of f i correspond to the same edge of Γ(T ). If two vertices of f i correspond to the same vertex of Γ(T ), then the crossing number of the projection can be reduced by an isotopy fixed on the boundary. Finally, if v j is incident to f i, then j {i, i + 1} or a crossing can be reduced by a free isotopy. Proof. Assume that two edges of f i correspond to the same edge of Γ(T ) (Figure 4.1). Then there would be a circle in the interior of the tangle circle intersecting the edge once. Then a string of T intersects this circle exactly once contradicting the Jordan curve theorem. f i f i Figure 4.1 Figure 4.2 Similarly, if two vertices of f i correspond to the same vertex, v, of Γ(T ), there would be a circle intersecting Γ(T ) only in v (Figure 4.2). This would give rise to a crossing that could be reduced by an isotopy rel B 3. Now assume v j is incident to f i with j i, i + 1 (Figure 4.3). Then e j / f i for otherwise T would be split. Thus e j lies in the exterior of f i. e j e j v j e i f i ei+1 f i Figure 4.3 Figure

33 But now we can untwist the crossing at v j by bringing e j into f i (Figure 4.4). Lemma 4.4. If T can be freely isotoped so that it has a projection with 7 crossings, then either T is split or has two parallel strands. Proof. Note that T being split or having parallel strands is invariant under free isotopy. Freely isotope T to a minimal projection with 7 crossings and let Γ(T ) be the corresponding graph. Assume T is not split. By Lemma 4.2, Γ(T ) has at least 6 vertices (thus T has at least 6 crossings). If the six v i are the only vertices incident to i f i, then (Lemma 4.3) we must have Figure 4.5. f 1 v Figure 4.5 Figure 4.6 But then T has a closed curve, a contradiction. Thus Γ(T ) must have another vertex v lying on f 1, say, which is not a v i. Since T has at most 7 crossings, there is exactly one such v and we have Figure 4.6. Enumerating the possibilities, we see there are nine cases, six of which are shown in Figure 4.7, while the remaining three can be obtained from the top three via reflection: Figure

34 Note in each case we see two parallel strands (since we assumed T is not split). For example, see figure 4.8. = = free free isotopy isotopy = parallel strands Figure 4.8 Proposition 4.5. Let T be a solution tangle which can be freely isotoped to a projection with 7 crossings. Then T is the PJH tangle. Proof. By Lemma 4.4, T is split or has two parallel strands. By Corollary 3.8, T is the PJH tangle. 5. T has at least 10 crossings Let T be a solution tangle. In this section we show that if T is not the PJH tangle (Figure 2.4), then T must be complicated, as measured by its minimal crossing number: Proposition 5.1. Let T be a solution tangle. If T has a projection with fewer than 10 crossings, then T is the PJH tangle. Remark. As pointed out in section 2, one goes from a solution tangle in normal framing to one in the [PJH] framing by adding one left-handed twist at c 2. Thus it formally follows from Proposition 5.1 that if a solution tangle under the PJH framing has fewer than 9 crossings, it must be the [PJH] solution of Figure 0.1. Here we are interested in crossing number up to tangle equivalence, that is, up to isotopy of the tangle fixed on the boundary. So we must fix a framing for our solution tangle. We take that of proposition 2.4, the normal form of our solution tangle. In this section, we focus on the fact that T s 23 is the 1/2 tangle. We use Proposition 4.5 (hence Corollary 3.8) and the fact that s ij s ik 2. Hence this section holds 34

35 under normal framing when the in trans deletion product is the (2,2) torus link while the remaining products are (2, L i ) torus links where L i 4. We begin with the following observation. Lemma 5.2. Let T be a solution tangle. In any projection of T, s ij s ik 2. Proof. Assume not. Then s ij s ik = 0. But then lk(ŝ k ˆx k, s ij x ij ) = lk(s jk x jk, s ij x ij ) = 0 contradicting Lemma 2.2. Let T be a solution tangle with crossing number less than 10. We assume that s 23 is the enhancer strand. To simplify notation we set e = s 23, α 2 = s 12, α 3 = s 13. Recall that T e = α 2 α 3 is the 1/2-tangle. Assumption 5.3. T has a projection, in normal form (see the beginning of Section 2), with at most nine crossings. Furthermore, among all such projections, take α 2 α 3 to have the fewest crossings. Remark. We will often blur the distinction between e, α 2, α 3 and their projections. We will first prove in Theorem 5.5 that Proposition 5.1 holds when α 2 α 3 has exactly two crossings. In this case let R 1, R 2, R 3 be the closures of the complementary regions of α 2 α 3 in Figure 5.1. Note all figures in this section have been rotated by 90 degrees so that the T e = 1/2 tangle will instead be displayed as an integral 2 tangle. α 2 R 1 e R 2 R 3 α 3 Figure 5.1 Lemma 5.4. If e crosses R 1 (α 2 α 3 ) fewer than four times then T can be freely isotoped to eliminate two crossings. Furthermore if e R 1 contains only four crossings of T then we can reduce T by two crossings unless e R 1 is as pictured in Figure 5.2(up to symmetry). 35

36 (a) (b) (c) Figure 5.2 Proof. Assume e intersects R 1 (α 2 α 3 ) in fewer than four crossings. By Lemma 5.2, e must cross R 1 (α 2 α 3 ) exactly twice. Then R 1 writes T c 1 as a disk sum. As T c 1 is a rational 2-string tangle, e R 1 must be an integral summand [L, ES1]. But then we can reduce the crossing number by at least two under a free isotopy by removing the crossings in the integral tangle e R 1 as well as the two crossings where e intersects R 1 (α 2 α 3 ). See Figure 5.3. Figure 5.3 Thus we assume e R 1 must have at least 4 crossings in R 1 with α 2 α 3. Assume these are the only crossings of T in R 1. Then there are no crossings in int R 1. This allows for two crossing reductions except in the cases pictured in Figure 5.2. See Figure 5.4. (Lemma 5.4) Figure 5.4 Theorem 5.5. Let T be a solution tangle satisfying Assumption 5.3. Under this projection, assume α 2 α 3 has exactly two crossings. Then T can be freely isotoped to have at most seven crossings. 36

37 Proof. Let R 1, R 2, R 3 be the closures of the complementary regions of α 2 α 3 in Figure 5.1. We divide the proof of Theorem 5.5 into two cases. Case I: e has 4 intersections in R 1. Case II: e has at least 5 intersections in R 1. Proof of Case I: R 1 contains exactly 4 crossings (involving e). In this case, there are 3 possible configurations up to symmetry (shown in Figure 5.2) that do not immediately allow a crossing reduction by two in R 1. Since there are at most 9 crossings, if e intersects R 2 then e R 2 must contribute exactly 2 crossings and e R 3 must be empty. But this allows a reduction of two crossings: In case (a) and (b) in Figure 5.2, the two crossing involving α 2 and α 3 can be removed as shown in figure 5.5. Figure 5.5 In case (c) one reduction comes from α 1 α 2, the other from e (α 2 α 3 ). (Figure 5.6). or Figure 5.6 Thus we take e disjoint from R 2, and we must have (a) or (b) in Figure 5.7 (a) (b) Figure 5.7 But (a) gives two crossing reductions and (b) contradicts Lemma 5.2. Q.E.D. (Case I) 37