# A Topological Characterization Of Knots and Links Arising From Site-Specific Recombination

## Abstract.

We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or -torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly with the cube of the minimum number of crossings. Additionally, we prove that the only possible products of an unknot substrate are either clasp knots and links or -torus knots and links. Finally, in the (common) case of -torus knot or link substrates whose products have minimal crossing number , we prove that the types of products are tightly prescribed, and use this to examine previously uncharacterized experimental data.

## 1. Introduction

Molecular biologists are interested in DNA knots and links, because they have been implicated in a number of cellular processes. The axis of DNA molecules can become knotted or linked as a result of many reactions, including replication and recombination. The wide variety of DNA knots and links observed has made separating and characterizing these molecules a critical issue. Experimentally, this is most conclusively accomplished via electron microscopy [15]. However, this is a laborious and difficult process. Thus topological techniques, such as those presented here, can aid experimentalists in characterizing DNA knots and links by restricting the types of knots or links that can arise in a particular context.

This work focuses on one such DNA knotting process, site-specific recombination, mediated by a protein, known as a site-specific recombinase. Site-specific recombination is important because of its key role in a wide variety of biological processes. (See [2] or e.g. [7] for more information). In addition, pharmaceutical and agricultural industries have become increasingly involved in genetically modifying organisms or testing whether a mutation in a particular gene leads to a disease. As a result, these industries are now interested in site-specific recombinases as tools for precisely manipulating DNA (e.g. [11]).

Site-specific recombination roughly has three stages. Two recombinase molecules first bind to each of two specific sites on one or two molecules of covalently closed circular DNA (known as the substrate) and then bring them close together. We shall refer to these DNA sites as the crossover sites. Next, the sites are cleaved, exchanged and resealed. The precise nature of this intermediary step is determined by which of the two recombinase subfamilies the particular protein belongs to (see Assumption 3 below for more details). And finally, the rearranged DNA, called the product, is released. Understanding precisely which knots and links arise during site-specific recombination can help understand the details of this process (e.g. [12]).

In this paper we begin by developing a model that predicts all possible knots and links which can arise as products of a single round of recombination, or multiple rounds of (processive) recombination, starting with substrate(s) consisting of an unknot, an unlink, or a -torus knot or link (denoted by ). This model rigorously develops and extends ideas that we originally sketched in [3]. Of all knots and links, we have chosen to focus on , because are the most commonly occurring knots and links in DNA . Our model rests on three assumptions that we justify biologically in [2]. Building on these assumptions, we use knot theoretic techniques to prove that all products fall within a single family, illustrated in Figure 10. We then prove that the number of product knots and links predicted by our model grows linearly with the cube of the minimal crossing number. We further prove that the product knot or link type is tightly prescribed when the substrate is and the product has minimal crossing number . Finally, we apply this new result to previously uncharacterized experimental data.

This paper complements earlier work by Sumners, Ernst, Cozzarelli and Spengler [18], which used the tangle model [9] and several biologically reasonable assumptions to solve tangle equations. They then determined which 4-plat knots and links arise as a result of (possibly processive) site-specific recombination on the unknot for the serine subfamily of recombinases (see just before Assumption 3 for a discussion of the two subfamilies). For the particular case of the recombinase Gin, they considered the knots or as well as unknotted substrates. Our paper goes further in several ways. In addition to allowing an unknotted substrate for a generic recombinase, we allow substrates that are unlinks with one site on each component, as well as any . Also, our assumptions are based exclusively on the biology of the recombination process. In particular, we do not assume the tangle model holds or that all products must be 4-plats. Allowing products which are not 4-plats is important because recombination has been seen to produce knots and links which are connected sums (see [2]).

We will use the following terminology and notation. Let denote a substrate which is either an unknot, an unlink, or (illustrated in Figure 4). We use the term recombinase complex, B, to refer to the convex hull of the four bound recombinase molecules together with the two crossover sites, and use the term recombinase-DNA complex to refer to together with the substrate . If the recombinase complex meets the substrate in precisely the two crossover sites then we say the recombinase complex is a productive synapse. In Figure 1, we illustrate two examples where the recombinase complex is a productive synapse, and one where is not. Finally, we let , and consider and separately.

The structure of the paper is as follows. In Section 2, we state our three assumptions about the recombinase-DNA complex, and use our assumptions to determine the pre-recombinant and post-recombinant forms of . In Section 3, we characterize the forms of for each of our substrates. In Section 4, we glue each of the post-recombinant forms of to each form of to determine all possible knotted or linked products predicted by our model. Finally in Section 5, we bound the size of this product family, and further limit product type in some special cases, by combining our model with results on minimal crossing number.

## 2. Our assumptions and

### 2.1. The three assumptions

We make the following three assumptions about the recombinase-DNA complex, which we state in both biological and mathematical terms. In [2] we provide experimental evidence showing that each of these assumptions is biologically reasonable.

(Biological) Assumption 1: The recombinase complex is a productive synapse, and there is a projection of the crossover sites which has at most one crossing between the sites and no crossings within a single site.

This is equivalent to:

(Mathematical) Assumption 1: consists of two arcs and there is a projection of which has at most one crossing between the two arcs, and no crossings within a single arc.

As a result of this assumption, we now fix a projection of such that has one of the forms illustrated in Figure 2.

(Biological) Assumption 2: The productive synapse does not pierce through a supercoil or a branch point in a nontrivial way. Also, no persistent knots are trapped in the branches of the DNA on the outside of the productive synapse.

Assumption 2 implies that the recombinase-DNA complex cannot resemble either of the illustrations in Figure 3.

In order to restate Assumption 2 mathematically, we first introduce some terminology. We shall use the term spanning surface to refer to a surface , bounded by , such that is topologically equivalent to a disk, two disjoint disks, or a twisted annulus when is an unknot, unlink, or , respectively. Figure 4 gives some examples of the relationship between a spanning surface and the productive synapse . Observe that in each of the illustrations of Figure 4, consists of two arcs. By Assumption 1, contains precisely two arcs of . Hence meets in precisely four points. It follows that the intersection of any spanning surface for with contains exactly two arcs. What we mean by does not pierce through a supercoil or a branch point in a nontrivial way is that does not pierce the interior of every spanning surface for (as in the left illustration in Figure 3). In general, a spanning surface is pierced by if and only if contains at least one circle in addition to the required two arcs. For example, in the diagram on the left in Figure 3, no matter how the spanning surface is chosen, contains at least one circle as well as two arcs.

Next, we explain what we mean by no persistent knots are trapped in the branches outside of . Consider a planar surface together with a finite number of arcs whose endpoints are on the boundary of the surface (see the illustration on the left in Figure 5). We can obtain a surface bounded by a knot or link by replacing a neighborhood of each arc in the original surface by a half-twisted band and removing the top and bottom ends of the band. Figure 5 illustrates how such a surface can be obtained from an annulus together with a collection of arcs defining the twists. Any surface obtained from a planar surface in this way is said to be a planar surface with twists.

Suppose that is a spanning surface for . We say that is unknotted rel , if there is an ambient isotopy of pointwise fixing which takes to a planar surface with twists, where the endpoints of the arcs defining the twists are disjoint from . For example, is unknotted rel for each of the spanning surfaces in Figure 4. This is not the case for the surfaces in Figure 3.

We now restate Assumption 2 mathematically as follows.

(Mathematical) Assumption 2: has a spanning surface such that consists of two arcs and is unknotted rel .

Site-specific recombinases fall into two families – the serine and tyrosine recombinases. Assumption 3 addresses the mechanism of recombination according to which subfamily the recombinase is in. While the overall reactions of the two families of recombinases are the same, they differ in their precise mechanism of cutting and rejoining DNA at the crossover sites. We explain more of the biological details in [2].

(Biological) Assumption 3: Serine recombinases perform recombination via the “subunit exchange mechanism.” This mechanism involves making two simultaneous (double-stranded) breaks in the sites, rotating opposites sites together by within the productive synapse and resealing opposite partners. In processive recombination, each recombination event is identical. After recombination mediated by a tyrosine recombinase, there is a projection of the crossover sites which has at most one crossing.

The mathematical restatement of Assumption 3 is almost identical to the biological statement.

(Mathematical) Assumption 3: Serine recombinases cut each of the crossover sites and add a crossing within between the cut arcs on different sites, then reseal. In processive recombination, all recombination events are identical. After recombination mediated by a tyrosine recombinase, there is a projection of the crossover sites which has at most one crossing.

### 2.2. The forms of

As a result of Assumption 1, we fixed a projection of such that has Form B1, B2, or B3 (illustrated in Figure 2). It follows from Assumption 3 that after recombination events with serine recombinases, we have added a row of identical crossings. Thus after recombination events our fixed projection of is isotopic fixing to one of the forms illustrated in Figure 6 (where the actual crossings can be positive, negative, or zero).

Also for tyrosine recombinases, we know from Assumption 3 that after recombination there exists a projection of with at most one crossing. We are working with the projection of which we fixed as a result of Assumption 1, and we cannot be sure that this particular projection of will have at most one crossing. However our projection must be ambient isotopic, fixing , to one of the forms illustrated in Figure 7. So without loss of generality we will assume that the post-recombinant projection of has one of these forms.

## 3. The possible forms of

Using Assumption 2, we now prove the following Lemma.

###### Lemma 1.

###### Proof.

We consider separate cases according to the knot or link type of .

Case 1: is the unknot.

In this case, by Assumption 2, we can choose a spanning surface which is a disk such that is two arcs and is unknotted rel . Since is a disk, the two arcs of separate such that one of and is a strip and the other is a pair of disjoint disks. Furthermore, if is a strip it is not knotted. Thus, is either a pair of disjoint disks or an unknotted twisted strip. It follows that is ambient isotopic, pointwise fixing , to Form C1 or Form C2.

Case 2: is the unlink

In this case, we assume that one site is on each component of (or else the substrate was actually an unknot). Thus by Assumption 2, we can choose a spanning surface which is a pair of disjoint disks such that meets each disk of in a single arc. Hence, and are each a pair of disjoint disks. It follows that is ambient isotopic, pointwise fixing , to Form C2.

Case 3:

In this case, by Assumption 2, we can choose a spanning surface to be a twisted annulus such that is two arcs and is unknotted rel . We see as follows that there are several ways the arcs of can lie in .

Any circle in must cross a sphere an even number of times (possibly zero). In particular, the circle representing the core of the twisted annulus must cross an even number of times. Each point where crosses is contained in . Since the total number of points in is even and consists of two arcs, either must intersect each of these two arcs an even number of times, or must intersect each of the two arcs an odd number of times. If each arc of intersects the core an odd number of times, then each of these arcs cuts into a strip. Hence the two arcs of together cut into a pair of strips. If each arc of intersects the core an even number of times, then each arc cuts off a disk from . In this case, either the two arcs cut off disjoint disks in , or one of the disks is contained inside of the other. In this latter case, the two arcs form the edges of a strip in , on one side of which is a disk and on the other side of which is a twisted annulus. The three forms of are illustrated on the top of Figure 9. Note that the illustration on the right may have one, rather than two, rows of twists. Since contains at most one crossing, the component of with almost all of the twists of must be contained in .

Since is unknotted rel , the abstract forms illustrated on the top of Figure 9 yield the corresponding forms of which are illustrated in the bottom of Figure 9 up to isotopy fixing . Observe that when has Form C4, the projection of must have Form B1 as illustrated. Also, in Form C3, while there may be twists to the left of , they are topologically insignificant, since they can be removed by rotating by some multiple of . Similarly, in Form C4, any twists which had occurred to the left of can be removed and added to the row of twists at the right by rotating by some multiple of . These rotations can occur while pointwise fixing .

Thus the four forms of illustrated in Figure 8 are the only ones possible. ∎

## 4. Product knots and links predicted by our model

In this section, we suppose that the substrate is an unknot, an unlink, or and that all three of our assumptions hold for a particular recombinase-DNA complex. Then we prove Theorems 1 and 2, which characterize all possible knotted or linked products brought about by tyrosine recombinases and serine recombinases respectively. If the substrate is an unknot or unlink we will also show that all nontrivial products are in the torus link family or the clasp link family (i.e., consisting of one row of crossings and a non-adjacent row of crossings). Note that is the well known family of twist knots and links. If the substrate is , then all products are in the family of knots and links illustrated in Figure 10.

Observe that in Figure 10, , , , and can be positive, negative, or zero. Furthermore, by letting , , , and/or equal 0 or 1 as appropriate, we obtain the five subfamilies illustrated in Figure 11. Subfamily 3 is the family of pretzel knots or links with three non-andjacent rows containing crossings, crossings, and crossings respectively. Observe that Subfamily 4 is a connected sum. However, if , , and , then it is a together with an unlinked trivial component.

###### Theorem 1.

Suppose that Assumptions 1, 2, and 3 hold for a particular tyrosine recombinase-DNA complex. If the substrate is an unknot then the only nontrivial products are or . If the substrate is an unlink, then the only nontrivial product is a Hopf link, . If the substrate is , then all of the products are contained in the family illustrated in Figure 10.

###### Proof.

We saw that as a result of Assumption 3, after recombination with tyrosine recombinases, the fixed projection of is ambient isotopic fixing to one of the five forms illustrated in Figure 7. Also, by Lemma 1, is ambient isotopic fixing to one of the four forms illustrated in Figure 8. For each of the four forms of , the products of recombination with tyrosine recombinases are obtained by replacing with each of the five post-recombinant forms of in Figure 7. The resulting products are illustrated in Figure 12. Recall that if is the unlink then must have Form C1, and if is an unknot then must have Form C1 or C2. The theorem follows from Figure 12.

∎

###### Theorem 2.

Suppose that Assumptions 1, 2, and 3 hold for a particular serine recombinase-DNA complex. If the substrate is an unknot then the only nontrivial products are or . If the substrate is an unlink, then the only nontrivial product is . If the substrate is then all products are in the family illustrated in Figure 10.

###### Proof.

We saw that as a result of Assumption 3, after recombination events with serine recombinases, the fixed projection of is ambient isotopic fixing to Form n1 or n2, illustrated in Figure 6. Also, by Lemma 1, is ambient isotopic fixing to one of the four forms illustrated in Figure 8. We obtain the products of serine recombinase from each of the forms of illustrated in Figure 8 by replacing with each of Form n1 and Form n2. The resulting products are illustrated in Figure 13. Note that when has Form C4, then must have Form B1. Hence the post-recombinant form of must be of Form n1. Recall again that if is an unlink, then must have Form C1, and if is an unknot then must have Form C1 or C2. The theorem follows from Figure 13. ∎

Table 1 summarizes the results of Theorems 1 and 2.

## 5. The minimal crossing number and our model

### 5.1. Our family grows with

The minimal crossing number of a DNA knot or link can be determined experimentally using gel electrophoresis. However, there are 1,701,936 knots with minimal crossing number less than or equal to 16 [13], and the number of knots and links with minimal crossing number grows exponentially as a function of [10]. By contrast, we will now prove that the total number of knots and links in our product family (Figure 10) grows linearly with . Note that, while the knots and links in our family have at most four rows containing , , , and signed crossings respectively, it does not follow that the minimal crossing number of such a knot or link is . If the knot or link is not alternating, it is quite possible that the number of crossings can be significantly reduced. Thus there is no reason to a priori believe that the number of knots and links in our product family should grow linearly with .

We begin with some results about minimal crossing number which will be used in our proof. We shall denote the minimal crossing number of a knot or link by MCN.

###### Lemma 2.

Let and . Then is equivalent to both and . Furthermore, if and have the same sign then , and if and have opposite sign then .

###### Proof.

Figure 14, we show that is ambient isotopic to by moving the highlighted strand in front of the diagram and then turning the horizontal row of half-twists so that they become vertical. Analogously, by moving the highlighted strand behind rather than in front of the rest of the diagram, we see that is also ambient isotopic to .

We evaluate as follows. Murasugi [17] and Thistlethwaite [20] proved that any reduced alternating diagram has a minimal number of crossings. Observe that if and have opposite signs, then the standard diagram of is reduced and alternating. In this case, . If and have the same sign and , , then either , or , . If , , then the diagram of is reduced and alternating, since all three rows of crossings are positive. In this case, . If , , then the diagram of is reduced and alternating, since all three rows of crossings are negative. In this case, . ∎

To prove our theorem, we will also make use of the following theorem of Lickorish and Thistlethwaite.

Theorem [16] Suppose that a knot or link has a projection as in Figure 15 with , and for each , is a reduced alternating projection which contains a crossing between the two arcs at the bottom of (as in Figure 15) and at least one other crossing. Then the projection of has a minimal number of crossings.

We shall adopt the language of Lickorish and Thistlethwaite and refer to a projection of the form described by their theorem as a reduced Montesinos diagram. Thus by the theorem, any projection of a knot or link which is a reduced Montesinos diagram has a minimal number of crossings.

###### Theorem 3.

The number of distinct knots and links in the product family illustrated in Figure 10 which have MCN grows linearly with .

###### Proof.

We begin by fixing , and suppose that is a knot or link projection in the family of Figure 10 which has MCN. This projection has crossings. If the given projection of is reduced alternating or reduced Montesinos, then . Otherwise, we show that is ambient isotopic to one of 24 possible projections which have a minimal number of crossings. We will then show that there are at most possible knots and links in our family with MCN.

The following example illustrates the type of strand move we shall use to reduce the number of crossings whenever the diagram is neither reduced alternating nor reduced Montesinos. Observe that the part of our knot or link consisting of the rows containing and crossings is alternating if and only if and have opposite signs. If and have the same sign, then by moving a single strand (as in Figure 16), this part of the knot or link becomes alternating. This isotopy removes a crossing from both the row and the row and adds a single new crossing. Thus we reduce this part of the diagram from having crossings in a non-alternating form to having crossings in an alternating form. All of the isotopies we use to to get rid of unnecessary crossings involve moving at most three such strands.

Next we will discuss the one exceptional case where we cannot obtain a reduced alternating or reduced Montesinos diagram by moving some strands of . This is the case when is a knot or link in our family with , , , and . In its original form, the projection has crossings. We can move a single strand of the diagram to obtain a projection with only crossings (illustrated on the left in Figure 17). We define a Hara-Yamamoto projection as one in which there is a row of at least two crossings which has the property that if this row is cut off from the rest of the projection and the endpoints are resealed in the two natural ways, then both resulting projections are reduced alternating. The projection on the left of Figure 17 is Hara-Yamamoto because the projections (on the right) obtained by resealing the endpoints are both reduced alternating. Hara and Yamamoto [14] have shown that any Hara-Yamamoto projection has a minimum number of crossings. Thus the projection on the left of Figure 17 has a minimal number of crossings.

We will consider 27 cases according to the values of , , , and , and show that in all but the above exceptional case is isotopic to a diagram that is either reduced alternating or reduced Montesinos and hence has minimal crossing number. Since there are so many cases, we display the results in a table rather than discussing each case individually. We make the following notes about the table. In the second column we list the form of the knot or link which has a minimal number of crossings (e.g. reduced alternating). If the knot or link is isotopic to a clasp, pretzel, or torus knot or link we will list the specific form (e.g. ). If the minimal crossing form is either a clasp or a pretzel of the form then (according to Lemma 2) which one of these is the minimal crossing form depends on the signs of and . In this case, we just list one of these two forms though the one we list is not necessarily the form with the fewest number of crossings, as we do not know the signs or specific values of the variables. In this case, for the MCN we write an expression with (-1?) at the end to mean that depending on the relevant variables the MCN may be one smaller. If one of these knots or links contains a trivial component, we use the shorthand O to indicate this in the table. We shall consider a knot or link and its mirror image to be of the same link type, and hence we will not count both. Thus without loss of generality, we shall assume that . Also, observe that the rows of crossings containing and crossings are intechangeable in Figure 10, so we treat the variables and as interchangeable. We list the MCN as an unsimplified function of , , , and to help the reader recreate the isotopy taking the original form to the minimal crossing form. Finally, apart from the cases where reduces to or , we obtain the upper bounds for the number of links in each case by expressing MCN as a sum of nonnegative integers. This enables us to find an upper bound for the number of knots and links with MCN in each case. Note that the upper bounds given are intended to be simple rather than as small as possible. In particular, a number of our cases overlap, and thus some knots and links are counted more than once. Also, for certain specific values of , , , and , we may obtain a trivial knot or link. However, we do not specifically exclude these cases from our table.

Values of , , , | Minimal crossing form | Strands | MCN written as a sum | Upper bound |
---|---|---|---|---|

for | moved | of nonnegative integers | on # of links | |

O | 0 | (-1?) | ||

0 | 1 | |||

, , | 0 | |||

, , | 0 | (-1?) | ||

, , | 0 | (-1?) | ||

, | 0 | 1 | ||

, , | 0 | (-1?) | 2 | |

, , , | reduced alternating | 1 | ||

, , | reduced alternating | 1 | ||

, , , | reduced alternating | 2 | ||

, , , | reduced alternating | 0 | ||

, , , | reduced alternating | 2 | ||

, , , | reduced alternating | 1 | ||

, , , | 0 | (-1?) | ||

, | 1 | 1 | ||

, , , | reduced alternating | 1 | ||

, , , | reduced alternating | 1 | ||

, , , | reduced alternating | 1 | ||

, , , | reduced alternating | 1 | ||

, , , | reduced alternating | 2 | ||

, , , | reduced alternating | 1 | ||

, , , | trivial | 2 | 0 | |

, , , | reduced alternating | 3 | ||

, , , | reduced Montesinos | 0 | ||

, , , | reduced Montesinos | 1 | ||

, , , | 1 | |||

, , , | Hara-Yamamoto | 1 |

There are 26 non-trivial cases in the table. However, all three instances of a yield the same knot or link. Thus there are at most 24 distinct families of knots and links listed in the table. The number of knots and link in each of these families is bounded above by (in fact, for most of the cases there are significantly fewer than knot and link types). It follows that for a given , the number of distinct knots and links in our product family (Figure 10 which have MCN is bounded above by . In particular, the number of distinct knots and links with the form of Figure 10 which have MCN grows linearly with .

∎

It follows from Theorem 3 that the proportion of all knots and links which are contained in our family decreases exponentially as increases. Thus, for a knotted or linked product, knowing the MCN and that it is constrained to this family allows us to significantly narrow the possibilities for its precise knot or link type. The model described herein thus provides an important step in characterizing DNA knots and links which arise as products of site-specific recombination.

### 5.2. Products whose MCN is one more than the substrate

Finally, we prove a more directly applicable theorem as follows. Site-specific recombination often adds a single crossing to the MCN of a knotted or linked substrate. If the substrate is and the product of a single recombination event has , then we can further restrict the resulting knot or link type.

###### Theorem 4.

Suppose that Assumptions 1, 2, and 3 hold for a particular recombinase-DNA complex with substrate , with . Let be the product of a single recombination event, and suppose that . Then is either , , or with and (see Figure 18).

###### Proof.

For , is the trefoil knot , and hence must be the figure eight knot which can also be written as . Thus from now on we assume that . By Assumption 1, there a projection of such that has at most one crossing. Since , the proof of Lemma 1 shows that is ambient isotopic, fixing to a projection with Form C2, C3, or C4 (see Figure 8). Furthermore, when has Form C4, then . By Assumption 3, after a single recombination event with either serine or tyrosine recombinases the post-recombinant form of is ambient isotopic fixing to one of those illustrated in Figure 7. Thus any knotted or linked product has one of the forms illustrated in Figure 12.

First suppose that has one of the forms illustrated when has Form C2 or C3. We see that cannot be , since MCN. Certainly, cannot be with a trivial component. If then , so we are done. If and , then , so again we are done. If and , then . In this case , and again we are done.

Now suppose that has one of the forms illustrated when has Form C4. If for some value of , then has a projection in the product family illustrated in Figure 10 with . Otherwise, is a member of the product family with . However, if , we can turn over the top loop to get (this will also add on positive crossing to the row). Thus we shall now assume that has a projection in the product family (i.e., Figure 10) with either or . Table 2 lists all of the nontrivial knots and links in this family when . Thus all of the products that we are considering occur in Table 2. We would like to know which of the cases in Table 2 have or , , and MCN. The following table answers this question.

Values of , , , | Minimal crossing form | MCN written as a sum | Is or | Can MCN |

for | of nonnegative integers | and ? | ? | |

O | (-1?) | no | - | |

no | - | |||

, , | yes | no | ||

, , | (-1?) | yes | if | |

, , | (-1?) | yes | no | |

, | no | - | ||

, , | (-1?) | yes | if | |

, , , | reduced alternating | yes | no | |

, , | reduced alternating | yes | no | |

, , , | reduced alternating | no | - | |

, , , | reduced alternating | no | - | |

, , , | reduced alternating | yes | only if | |

, , , | reduced alternating | yes | no | |

, , , | (-1?) | yes | yes | |

, | yes | yes | ||

, , , | reduced alternating | yes | no | |

, , , | reduced alternating | no | - | |

, , , | reduced alternating | yes | no | |

, , , | reduced alternating | yes | no | |

, , , | reduced alternating | no | - | |

, , , | reduced alternating | yes | no | |

, , , | trivial | yes | no | |

, , , | reduced alternating | yes | no | |

, , , | reduced Montesinos | yes | no | |

, , , | reduced Montesinos | yes | no | |

, , , | no | - | ||

, , , | Hara-Yamamoto | no | - |

The only subtle case in the table is where . In this case we must have , , and . It follows that . Since is non-trivial, we must have . Now MCN implies that . Thus where . Now from Table 3, we can see that if or , , and MCN, then is either , , or with , , and . ∎

We now illustrate an application of Theorem 3 (further applications of our model are discussed in [2]). Bath et al used the links and as the substrates for Xer recombination, yielding a knot with MCN=7 and a knot with MCN=9, respectively. These products have not been characterized beyond their minimal crossing number, and MCN is not sufficient to determine the knot type. In particular, there are seven knots with MCN=7 and 49 knots with MCN=9.

Theorem 4 significantly reduces the number of possibilities for each of these products. In particular, it follows from Theorem 4 that the 7-crossing products of Xer must be , or ; and the 9-crossing products of Xer must be , , or . All of these possible products are actually 4-plats. This example shows how our model complements the work of [8], which restricts attention to the tangle model, and thus assume that all products are 4-plats. In [1], building on earlier work of [21, 8, 4, 5], we use our model together with tangle calculus to completely classify all tangle solutions to these Int-Xer equations.

## 6. Acknowledgements

The authors wish to thank Andrzej Stasiak, De Witt Sumners, Alex Vologodskii, and Stu Whittington for helpful conversations. Dorothy Buck was partially supported by Grant # DMS-0102057 from the National Science Foundation’s Division of Mathematical Sciences. Erica Flapan was partially supported by an Association for Women in Mathematics Michler Collaborative Research Grant.

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