 Thank you the way. Do you all hear me well? Yeah, so thank you the way it and first of all I want to thank the organizers. It's really I'm really really happy to be back in Trieste last time I was here was eight years ago, and I had a four months old baby with me So this time I'm more relaxed and hopefully we'll get to see the city and Very happy that the city has welcomed us with such a beautiful weather. So that's just That background is the sunset yesterday This is my point that yeah Okay, so I'll tell you about pathways of unlinking by local reconnection. I'll start with a little with a little motivation Local reconnection events appear in many different contexts. For example, they appear as Magnetic reconnection in solar color coronal loops In noted vortices and fluid dynamics. So this is an experiment performed in Irvine slab where they 3d printed knots and two component links with different topologies They blew a fluid through it and there will be a few talks this week that will refer to this with much more expertise than the one I have and The bubbles that came out were knotted and these knots Slowly on knotted themselves or the link slowly on link themselves until they got to a trivial configuration and Locally each on linking reaction was mediated by a reconnection event that locally looks like this This is very very similar to Local reconnection that we see in the context of Recombination in biology and that's what brought me to that topic I mean to thinking about local reconnection In the global sense. So so my work for many years has been on the action of enzymes that change the topology of DNA So for example site specific recombination enzymes or type 2 topoisomerase enzymes I will focus in this talk on site specific recombination But a lot of the things that I'll say here are applicable to other reconnection events in nature And so in physics and chemistry So site what are site specific recombinases? These are enzymes that bind two short and identical Sequences of DNA and by short I mean between five and fifty base pairs long. So these these are very Steve pieces of DNA segments of double stranded DNA and they act by a cut recombine and paste reaction So they will introduce a double stranded break in each one of the two sides So here the sites are indicated by red arrows If you don't see one arrow is pointing from left to right and the other one points from right to left The blue balls are representing two specific enzymes So for example Xer C and D that I have worked on but it could be any other site specific recombination enzyme and in the case of Xer C and D these enzymes Belong to the tyrosine family of recombinases and they cleave the DNA in two steps Other enzymes in the serine family cleave the DNA in just one step They introduce double stranded break at once, but here we see a small cleavage a strand exchange Followed by a nice organization step where the second cleavage sites come together. There's a second cleavage Recombination event and the net effect is that you go from these configuration to these reconnected configuration Changes in the DNA mediated by these enzymes can have topological effects and that's what brings us here But they can also have very important phenotypic effects. So just to give you an example. I'm here. There's three major examples But let me just talk about this one the process of integration This is used by viruses for example bacteriophages viruses that infect bacteria will introduce their DNA So will will recognize the the cellular host and they will inject the DNA into the cellular host and if the virus is a lysogenic virus it goes into a lysogenic cycle by integrating its DNA into the host's DNA So in in this cartoon, this would be the viral DNA Circular viral DNA the DNA circularizes upon entering the cell and these on the left will be the bacterial DNA For example and the site specific recombination reaction for example integrates of lambda int will integrate the viral DNA into the bacterial DNA Which can then be replicated at every cycle and be inherited by any new daughter cell before being excised again and Generating new viruses to kill the cell. So that's just one example But there's other important examples for example inversion of segments which can change the genome and As it changes the genome as a by-product it might have a topological effect Okay, so so I'll focus on site specific recombination Let me rewind and get you started since this is a very interdisciplinary audience For people who are not working actively on on by in biology Let's remember that DNA is a right-handed double helix consisting of two sugar phosphate backbones Each backbone is lined up by nucleotides Adenine, timing, guanine and cytosine. Adenine pairs up with timing guanine pairs up with cytosine via hydrogen bonds and because the sugar phosphate nucleotide unit Is not symmetrical when they pair up as they stack on Double helix is formed. So this is a Cartoon of the right-handed double helix. This structure was proposed in a seminal paper by Watson and Crick in 1953 where they based Their model for the structure of DNA on the x-ray crystallographical work of Rosalind Franklin and Ryan Raymond Gosling who was her graduate student at the time and in particular on these specific photo 51 Okay, so Watson and Crick wrote a paper in nature in 1953 where they proposed this double helical structure of DNA and in that paper They have a sentence where they say well this structure of DNA automatically suggests a copying mechanism for the molecule so in later that year they published a second paper also in nature where they Started exploring this idea on how the DNA is replicated inside the cell and In that paper they write since the two chains in our model intertwine It's essential for them to untwist if they are to separate So during replication Helicases come in and they open up the helix they break the hydrogen bonds and open up the helix so that's the untwisting and Although it's difficult at the moment to see how these processes occur without everything getting tangled We do not feel that this objection will be insuperable and I show this quote a lot because Yes, indeed I mean if you have to if you have a helix and you open it the torsional stress from the helix will be Accumulated on one side you will have a huge accumulation of positive supercoils and depending on the initial geometry and topology of the of the molecule Well, this will result in everything getting tangled But the cell solves this problem every single time so How is the cell solving this problem? it's using enzymes called type 2 to boy summarizes and So before saying that let me focus on circular DNA molecules Here is a circular DNA molecule for example a DNA plasmid or a bacterial chromosome if we think of this green circle as a bacterial chromosome and these are the origin of replication here are two replication forks moving bidirectionally toward the termination region and As they move along ahead of each Replication fork you have an accumulation of positive supercoils Some of these positive supercoils are removed by type 2 to boy summarizes are relaxed, but others persist there some of them are diffused behind the replication fork forming Precutinase and at the end of replication there is an accumulation of replication links or replication cutinase so the quote that I put and these images are just to illustrate to you that The re the resulting replication cutinase are a direct Consequence of the double helical nature of DNA. I mean there's no mysteries in there You have a double helix you open it up if the helix is circular you will get Are you still hearing me already? Yeah, you will get two component links and These two component links well will pose a problem during segregation of chromosomes at cell division because now each one of these Components is an identical copy of the original Parental DNA molecule and they need to be segregated to separate cells, but they are topologically linked so the The cell needs to solve this problem and it solves it using type 2 topoisomerases in the case of the bacteria Micoly the type 2 topoisomerase that is believed to be in charge of this decatenation is top of 4 and top of 4 will go in and By strand passage reaction will simplify the topology of DNA There is experimental evidence that the other enzymes can also assist with the decatenation process top of 4 is Probably most certainly the most efficient one so type 2 type 2 topoisomerases are very fast and very efficient and Fast and efficient don't they don't mean the same thing for me So fast means the reaction is very quick and efficient means the reaction really follows topological pathways that are optimal for Unlinking so the type 2 topoisomerases know how to do this very well But there's other enzymes that also seem to be assisting in the process like type 1 topoisomerases top of some top of 3s have been found to do this and The case that I'm going to tell you about so I Collaborate with Dave Sherrod in Oxford and in the Sherrod lab in 2003. They generated these type of DNA links these are called torus links to and torus links They generated two and torus links and they generated torus knots with lambda and with a site-specific recombination enzyme and They treated that with Xcrcd the fft sk so Xcrc and the are site-specific recombinances They are the two sites that are recognized as the name of the sites and ftsk Is a powerful transfer case that is believed to play an important role in the cell to bring the sites together and When they incubated these links with Xcrcd, they saw that this complex was able to Do the un-linking slowly, but surely the complex was able to do the un-linking so that raised the question Is this happening in vivo? Is this happening with replication links? These were just linked plasmids So they went back to living cells. They extracted Linked plasmids tied up During replication in vivo and they show that the complex can also unlink them in vitro And they also have evidence that the comp these complex can unlink the replication links in vivo So so that was published in this paper all the experimental evidence and we also published So there's a little mathematical section in that paper where we publish How we thought this was happening so what the pathway what we thought the pathway was and The topological mechanism with a very basic topological analysis. So I've continued this work For the last few years Okay, so this is joint work. I already mentioned Dave Sherrod at the University of Oxford For that paper the 2007 NEMBO paper Ian Grange did a lot of the experiments He was a postdoc in Dave's lab at that time and now he's a professor at the University of Newcastle in Australia And on the mathematical side, I collaborate with Koya Shimokawa from Saitama University And the work I'm going to present is work of his master student Masaki Yoshida and Kai Ishihara who was Koya's PhD student when we were doing this work Kai is now a professor at Yamaguchi University in Japan Okay, so there's two key questions The first question and this is the one that I will focus on right here. It's a question on Reconnection pathways what are so the question is the following what are the shortest topological pathways? from a given replication link to the unlink so I can get a link with two four six eight any two and crossings and Assuming that the local reconnection reaction unlinks this link How does it do it? What pathway does it follow and at each step of the pathway? That's the next question What is a topological mechanism? So at each step of the reaction if I go from a six crossing tourist link for example to some not our link with Five six or five crossings or whatever number of crossings. What is the mechanism at this level? Okay, so for this we need a mathematical model of recon of local reconnection This is a completely topological model Some of you in the audience you have your mind completely wired up to physics there's no energies in this model and You should I mean we should talk to put the energies in it But this is completely topological mode. So DNA is simply modeled as a curve drawn by the axis of the double helix and If the DNA molecule is circular then this curve is automatically a circle and a circle that is properly embedded in three-dimensional space It doesn't self-intersect and That's the definition of a mathematical note and indeed DNA can be noted or interlinked. This is a note published in the Kosarelli lab 1985 these are two interlinked DNA circle circles published by the J.R.M. and the Harshi lab in 2002 And there's many examples like that So a circular DNA molecule will be modeled as a mathematical note and a union of Circular DNA molecules will be modeled as mathematical links as these joint unions of circles that are properly embedded in three-dimensional space now Here is a table of knots and I want to pause here a moment From and take a break from the biology to talk about nomenclature because I think nomenclature is very important for our community as we saw In in the talks this morning I mean they the speakers beautifully illustrated the importance of chirality in chemistry in physics And also in biology So and we also saw the original knot table by Tate and if we look at that knot table And at any knot table in the back of a knot theory book So most not your books have a knot table in the very back There's no way to tell whether or not is let's put it in quotation marks positive or negative And I would like to have a way to distinguish between What I will call positive knots and negative knots so that I have an unambiguous way and unambiguous nomenclature When I'm doing any topological analysis of reconnection for example so in we did an extensive study of rise a few years ago and I should have put that other paper here. So we pop we publish a paper. I think it's 2011 and then we publish this small paper two thousand in 2013 where we're proposing to you that our community uses a nomenclature that is guided by the rise and This is based on the conjecture that if we have a knot and We consider the space rise of that not so the rise of the knot if we just look at a not diagram So for example here we assign an orientation to the knot diagram And then we look at the at the sign of each one of the crossings We add the sum of the cross we add the the crossings and for these travel for example We get plus three. Okay, so that's the projected rise now If I take that to three dimensions the average of the projected rise over all directions can be described as a gauss-double integral and It's a geometrical invariant. It's not a topological invariance So for the travel if the travel is close to its ideal form the average That's the mean right or the average right will be around three as well three point two or three point one depending on the length of the knot and all that and now Let's consider an ensemble of trefoils so all possible conformations of the travel with all possible lengths and Let's take the average of all those rides that we will call that the overall right and This these concepts that we didn't introduce these concepts many people have worked on them and The conjecture is that the sign of the overall rise of a chiral knot is a topological environment that you can always Distinguish a chiral knot by the sign of the overall rise of course computing this is not so easy So our study was a numerical study where we approximated this overall rise with computer simulations of Polygonal chains both in the lattice and off lattice and for all knots with up to 10 crossings We were able to separate that average Overall that overall ride from zero with the error bars keeping it away from zero and Determine what the sign was if the sign was positive We put that knot in the table if the sign was negative then it's the mirror image So we have the table here It's with up to 8 crossings But if anyone wants to use the table this is with up to 10 crossings the table if we remove the color is identical to any knot table in any notebook the difference here is So if if we look at the ropes and stable, which is the standard The trifle in Rawson's table is negative So if my notice is red that means that I have the opposite the corollity and that of Rawson's And if the knot is yellow, then it's the same corollity as out of Rawson's or the knot is a carol in which case it doesn't matter Okay, so so there's a table and nomenclature is important now nomenclature for catenase for links is also extremely important. Yes Yeah, yeah, they all they they're all Positive even even in the minimal projection I mean in some cases you have knots where the projection of the figure in the table has right zero But but in this case they should all have positive right if they're on this table the same sign That's forever. I guess the limitation would be at the beginning But the first one well the first one is this one, but I mean here and then the first one that is this one Is a carol so that's not a problem So the the right is here Okay, so for links this illustrates the problem for links for links We have not only to consider the mirror images But we also have to consider relative orientation of the side of the of the components So so if I call these the link that will be in my link table, then I can consider It's mirror image and then I can reverse the orientation of the original one of One component of the original one and I can reverse the orientation of one component in the mirror image And I could also consider a case where the two green arrows point from right to left instead of from left to right And that mathematically can be considered as a completely different link. We choose not to do that. We just Allow four forms for each two component link, but I mean the the nomenclature Deciding which link goes on the table is difficult. So and and why is it important? So for example in the context of reconnection as you will see later if I have these Link with these chirality and this orientation with one reconnection event I can go to the 5 1 6 2 or trefoil if I instead of taking this one I take this one right here. I go to completely different topologies So it is very important that we have an unambiguous nomenclature as we are now We go into the not theory tables in books We go into the not theory tables available online like not not in for not plot and this is quite ambiguous So that makes the the details of the work difficult if you're trying to work on on not and links Okay, so so we propose a nomenclature for for the small ones that we're using and well This is something we're working on and the comparison with the others Okay, and in our in our work will also see split links, which are just these joint unions but also that that were the linking number is zero and Connected sums like this one or like this one Okay, so let's go back to the biology so it's specific recombination as I explained before will introduce in one or two steps a double stranded break reconnect and Receal the break with this net effect and In the case of site specific recombination this reaction at the biochemical level is very well understood So we really understand what are the amino acids here that that Cleave the DNA and everything that's happening inside here. It's pretty well understood But that doesn't help me to understand large topological changes now these enzymes can or not and on link DNA as I motivated at the beginning and The question will be how are the enzymes inducing these large topological changes? Yes, locally we understand how they cleave and perform trying to exchange But globally we don't necessarily understand and the local action does not Imply a global action in general So this is illustrated here where we have a circle It could be a DNA plasmid with two recombination sites with a particular orientation And then this circle is moving in solution. It's a supercoil DNA The two sites are juxtaposed here with trapping three crossings in the domain and reconnection Produces a two-component link a hop flink non-trivial link Whereas if only one crossing is trapped reconnection produces a non link so what we conclude for this from this example is that the product topology is a direct consequence of the Geometrical conformation adopted by the substrate prior to local reconnection even if we understand what's happening at the local reconnection level The global geometry will be crucial to understand the whole process Okay, so we model the reconnection as a two-step reaction where the substrate In the case of these talk the substrate will have complicated topology because we're interested in on linking DNA so we're interested in topological links or knots but Here in this example, we only have a non-knotted DNA molecule it goes into a black box where the local the binding happens and the local reconnection event takes place and You obtain either a unique product if the enzymes have topological Specificity or a range of topologies a range of product topologies Which can be knots or links if we look inside under the microscope under the electron microscope This is what we see. I mean, this is not my image. This is an old image of Hicksman in In the 1990s, I think or 1980s and Here there's site specific recombination enzymes attached to DNA and you see two emanating loops of DNA So these loops here are double-stranded DNA are pieces of the double-stranded DNA molecule If we started with a single Circle as a substrate and we see this if we see two loops coming out There must be two loops inside and this is part that with some nurse here in the audience and he stands to the Clause Ernst to propose the tango method to study such specific Recombination so they said, okay. Well, let's model this enzymatic complex bound to the DNA as a two-string tango which mathematically is a three-dimensional ball, but it's a topological ball so it can be Deformed like illustrated in this figure with two properly embedded segments inside that intersect the ball in in four distinguished points a b c and d and We go even further to add a framing to this tangle where and that framing is In the form of a homeomorphism of pairs that takes a three-dimensional ball into the unit ball in three-dimensional space See there are three or s3 and the four points a b c d two four points on the equatorial circle north west northeast south east south west and Tangles can be studied through their tangled diagrams just like knots and links so one can project on to the XY plane and obtain a tangled diagram and Most tangled diagrams will be regular so one can distinguish between over and under crossings unambiguously Okay, so so this is a definition of two-string tango level two So two two strings two segments that are intertwined in some non-trivial way, maybe or in a trivial way and a homeomorphism of pairs that anchors the tango and these homeomorphism of pairs makes life easier, but it also adds complications and Two-string tangos can belong to three families. They can be rational locally knotted or prime a locally knotted tangle is one where you can find a Local knot in one of the strands and you can isolate it with a sphere around it that intersects the strand in only two points a Tangle is rational if it's homeomorphic to a trivial tangle This is a trivial tangle right here for example, so if you can deform this tangle without breaking the chain and Turn it into something different. What what you obtain is a rational tangle and the prime tangos are all the others and And most tangos in biology are Rational or sums of rational tangos So as I mentioned you start with a trivial tangle with two straight arcs or just one crossing So the trivial tangos are this one with the two vertical straight arcs or that's called the infinity tangle or the zero tangle which is a Tangle like this with two horizontal arcs or the plus or minus one tangle where you have only one cross in either orientation in your chirality and You can take this ball with two straight arcs and With moves similar to the rubik cubes moves You can have to do horizontal twists and vertical twists, but the finite sequence of such moves you can tie any possible rational tangle and This is thanks to the classification theorem by Conway in 1970 Which puts the rational tangos into a one-to-one correspondence with the extended rational numbers the Infinity corresponds to this tangle right here and any other rational tangle Can be uniquely labeled with a rational number and from the rational number with a continued fraction calculation We can extract a recipe from for tying this tangle from the trivial tangle So for example here, you're taking the trivial tangle and you're doing negative five Vertical twists followed by negative two horizontal twists and the negative and the positive is just a sign convention Okay, so so now that we know about tangles Let's talk about the tangle method in the tangle method. There's some assumptions. So the first one is that the Enzymatic complex is modeled as a two-string tangle But these two-string tangle we're going to compartmentalize it we're going to divide it or partition it into two compartments O and P this division is called the tangle sum So the time the enzyme tangle E is written as the sum of O plus P and P I mean this figure is a bit misleading because O and P seem to have the same size In reality P will be a really tiny tangle that only contains the cleavage region It contains a very very short piece of DNA that is not flexible at all so the p-tangle will be a trivial tangle and All the old tangle contains any other complicated topology outside of P which will determine the topology of the product after recombination and the next Assumption is that recombination goes by tangle surgery where it replaces a tangle P with another trivial tangle R and I mean R could be non-trivial But if we think in terms of DNA and how short the DNA molecule is in this tangle The tangle R will also be a very simple tangle Okay, and then throughout the reaction all remains constant nothing happens inside O We go down to our assumption. So for the the biological application Here is in the case of Xer-C and Xer-D. Here is a map of the recombination site It's a very very short sequence of DNA consisting of 28 base pairs and Considering that the persistence length of DNA is 150 base pairs 28 base pairs is very stiff Very hard to bend so we can safely assume that P is a trivial tangle and since we're setting up the model This is the first thing we do when we start the model We're going to position ourselves so that P looks like two straight arcs in a horizontal position Now each of the recombination sites the recombination sites are described by a sequence of very specific Sequence and this sequence is typically non-palindromic in some experiment It's been made palindromic, but it's typically non palindromic So one can assign an orientation to each recombination sites the two sites are identical So if we think of P as a zero tangle when we plot it We can see the recombination sites as pointing in the same direction here from left to right or as pointing in opposite directions Here we say that they are in parallel alignment that the sites are in sorry here in parallel alignment And here we say that they're in anti-parallel alignment and recombination goes through these two steps through a college at junction intermediate In the case of anti of O anti-parallel R will be the infinity tangle in the case of O parallel R will be the plus one Tangle or the minus one tangle and this is solely based on The knowledge of the biochemistry that we have at this level for other applications of reconnection the assumptions might be different Okay, so now for each substrate the action is translated into a system of tangle equations a system of two tangle equations Where this green this blue tangle is the enzyme tangle that I showed before and the construction that closes The loops at the top and the bottom so north east to northwest and south east to southwest is called the numerator operation So this is called numerator of E, but E is O plus P So this is called numerator of O plus P before recombination and it's called numerator of O plus R after recombination And before recombination we have a substrate topology with a very specific topology K naught and after recombination We have a product for a single recombination event We have a single topology product if we have recombination on an ensemble of substrates then the products my vary the topology of the products may vary depending on the enzyme that we're working with and O P and R are two string tangles We know that P and R are simple, but we don't know exactly what R looks like and oh we have no idea what it looks like so we want to solve the system of equations for the tangle O and For for R and we know it's going to be small But which one of the trivial tangles is it so the tangle equations There's a lot of things that are known about tangle equations and tangle theory in the mathematics community so if oh and We know that if we take the sum of two rational tangles the numerator of the sum of two rational tangles is In the family of four plots or two bridge knots and two bridge knots have also been classified So that led Ernest and Sumner's to Developing tangle calculus which allows us if we know that the tangles involved in the reaction are rational or sum of rational tangles the tangle calculus allows us to find solutions to the tangle equations and by translating the three-dimensional image of the tangle to a Very sim very simple algebraic equations that come by virtue of the classification of rational tangles There's software to to solve rational tangles to to so if you're interested to just You can plug in the not the substrate the product and see what the different topological mechanisms are This is a software that is a stand stand alone a Java applet But there's also a part of not plot developed by Isabel Darcy with Rob Charin within not plot that also solves tangle equations And they're not identical. So theirs has more things and we do and ours has Contemplates other aspects. So it's worth looking at both of them if you're interested in tangles, okay So we get a system of tangles equations. We want to solve it for O and R and These leads to the collaboration with low-dimensional topologists and more And not theorists because these solving tangles equations poses interesting problems I said we know how to solve the equations if the tangle all is a rational tangle or a sum of rational tangles But what if it's locally noted? What if it's prime? Then we don't necessarily we don't have a tangle calculus and we don't necessarily know how to solve the tangle equations So in the best case scenario if I have a system of tangle equations arising from a from biology I would like to prove mathematically using tools from then surgery that all is a Rational tangle or a sum of rational tangles and then I can rely solely on the on the tangle calculus This is not always possible But but that's and I'll go fast through this because I have almost no time But I can talk to you about this if you want and I know Dorothy will probably talk a lot about this on Thursday Okay, so So I'll I'll I'll skip forward to the reconnection pathways, which is really what I want to tell you about today So again, what are the different ways of going from a replication link to the on link? We needed all this mathematical formalism for the results that we have about pathways so the first result we have is that the shortest on linking pathway of Any 2m catenaine? So this is not restricted to the six crossing link or eight crossing links so if you have any 2m crossing torus link of This type right here. I mean that looks like that then the shortest on linking pathway has exactly 2m steps So if you start with a eight crossing link then the on linking pathway has to have eight steps These links are the replication links and they have a very specific orientation of the sites if the orientation is Reverse then you can go to the on linking two steps Okay, so that's the first result and And the first result just relies on modeling these with with two string tangles using the tangle method assumptions Okay for the next result we need to ask something so now we want to know okay So the shortest pathway to on link an eight crossing torus link has eight steps How many shortest pathways are there? Is there just one? Can I find one? So can I find one the answer is yes This is the obvious one and this is the one we publish in the biology paper But are there any others this is very very important and interesting are there any others to show? Whether there's there's others or not with we need to start by making some assumptions so we look at The gel electrophoresis coming from the experiments and we see that the complexity of the products goes slowly down until The topological links are on linked So based on those experiments we say okay Well, let's assume that at each step of the reaction the crossing number goes Strictly down if we assume that the crossing number goes strictly down So I start with eight crossings and then I go to seven or fewer then We can show that if I start with an eight crossing link or a six crossing link So let's say this is an eight crossing link Then the next step will be a seven one knot if I start with a six to one link Then the next one will be the five one knot and the pattern continues for all the two the the two end Torus links and the same thing can be shown with not four knots So that means that these six cut an end will go to the five one knot the five one not will go to the Four cut Which is a one we saw this morning that four cut will go to the trefoil etc Which is proving that the shortest on linking pathway Under that assumption is unique. So this is the only way This is the only pathway to take you to these two and torus link to the on link Okay, well That's the unique pathway under a very stringent assumption So we wanted to remove the assumption which made us uncomfortable because we're mathematicians and the biological experiments were not really Conclusive about this in the trend was going down, but was it strictly going down? That's not very clear So instead of strictly there right here We knock that down and we say what if it goes down or stays the same if it goes down or stays the same At every step are there other pathways? The answer is yes, and we show that there's nine other pathways and In this case for the starting with a six crossing link There's all these nine pathways the one on the diagonal is the one that we had found before and Here there's a lot of low-dimensional topology work that has happened some In my collaboration with Koyashi Mokawa some that Dorothy and Kayeshihara have done and That work allows us to say that at every step here each of the red steps have been characterized So we can find the solutions using tangle calculus for the black steps. We still don't know the answer and Okay Well, but now there's nine pathways, and I want a unique pathway, right? I mean if you're looking at reconnection in biology and physics in chemistry You want to know are there many pathways or is there only one so the next question is which one is the most probable pathway? That's what I really want to know. So so then we did Numerical work and I'll zoom through this so our knots are simple cubic lattice knots and We perturb them we generate ensembles of conformations of knots using the BFACF algorithm which is ergodic within the knot types and We define a recombination procedure where we find Recombination sites along the chain. So here's a that white bubble is a recombination site And if there's only one we perform recombination on it if there's more than one we choose one at random and perform Recombination of it on it, which is our local reconnection event right here And by doing that we slowly can I mean we can explore the the different Products and their probabilities their transition probabilities from a specific topology to other topologies So here we started with the eight to one torus link and slowly now We're at two crossing link then there will be an on knot and and then on link So so that's just to illustrate the process and I'm almost done So the result for the limited graph But the numerical data is not assuming that the complexity remains or goes down The numerical data is allowing the knots and the links to go anywhere the complexity can go up But if we just restrict to this graph, then it's clear that the most favorite pathway is the one along the diagonal Which is exactly what we were hoping for and what we thought would happen these illustrates all the different pathways that we see with less complexity, I mean and starting with nine one and going down and this is a whole Breath of the numerical experiment. You're not supposed to understand this. There's no way to understand it But it's just to illustrate That the moment you start working with knots and links and start allowing reconnection to go up in complexity You can get to very complicated knots and links. Okay, so this is my group Thank you, which I have joined with a Javier Aswaga and Thank you to my collaborators and my funding. Thank you so much