Hydrocarbon Chains: Stuck Unknots? By Michael White

Can a hydrocarbon chain realize any stuck unknots? Overview This talk will focus on the link between knots, stuck unknots and realizing these knots as long strings of hydrocarbons. Can a hydrocarbon chain realize any stuck unknots?

What is a Knot? A knot is a closed curve in space that does not intersect itself anywhere. It has no thickness, its cross-section being a single point. Images From http://www.ualberta.ca/CNS/RESEARCH/3DPrinter/ImageGallery/knot.dir.1024x1004.jpg Images From: www.aperfectworld.org/ metaphors.htm

Deformations of Knots Deforming a knot does not change it. Topological knots are assumed to be made out of deformable rubber – like an extension cord or a rubber band. Image From: (Get from Mary)

Some Definitions Over Strand Under Strand Crossing Number – denoted c(K) Over strand: goes over at a crossing Under: goes under at a crossing Crossing number: The minimal amount of crossings in any projection of a knot Images from: www.knotplot.com

The Trefoil, Unknot, and Figure Eight Knots Crossing Number: 0 Crossing Number: 3 Crossing Number: 4 The unknot, or trivial knot, then the Trefoil knot, and the figure 8 knot. Images From: www.knotplot.com Images From: www.knotplot.com

Stick Knots Stick knots differ from the normal description of a knot in several ways. A regular knot has no thickness, and is allowed to be deformed in any way. Stick knots are not an infinitely bendable strand, but rather are allowed certain lengths, and certain angles, and the individual sticks can not be bent. This is the stick knot of the trefoil (s(K) = 6). The points with P next to them are lying in the plane of the screen, the points with B are behind the plane, and F is in front of the screen. This knot is not able to untwist itself into the unknot. These knots have several equations associated with them. Image From: http://pass.maths.org.uk/issue15/features/knots/

Stick Knot Equations S(K) = stick number of a knot These two equations put lower bounds on the stick number of a knot, the latter being a more precise bound. S(K) = stick number of a knot C(K) = crossing number of a knot Equations from: Negami and Calvo

Stuck Unknots Stuck unknots are a group of stick knots that are topologically unknotted or not even closed, but because of the restrictions on length and/or bond angle can not be realized as their topological equivalent. If you look carefully at the, you will realize that it is, in fact, unknotted. The only reason that it is not able to unravel is that the long sticks prevent the knot from being able to deform itself into the desired shape. (Heather) There was a theorem in Heather Johnston’s paper Non Trivial Embeddings of Polygonal Intervals and Unknots in 3-Space that stated that for this chain, as long as l1 and l5 were greater than or equal to l2+l3+l4, then this chain would be a stuck unknot. Images from: Non Trivial Embeddings of Polygonal Intervals and Unknots in 3-Space, Heather Johnston and Jason Cantarella, J.Knot Theory Ramifications 7 (1998), no. 8, 1027-1039. Images from: Non Trivial Embeddings of Polygonal Intervals and Unknots in 3-Space, Heather Johnston and Jason Cantarella, J.Knot Theory Ramifications 7 (1998), no. 8, 1027-1039.

Hydrocarbons Nomenclature Single bond – Alkane Double bond – Alkene Triple bond - Alkyne Hydrocarbons are simple molecules that are made only from carbon and hydrogen atoms. Carbon atoms like to have 4 things bonded to them, which, for a hydrocarbon, will only be other carbons or hydrogens. If a carbon atom is bonded to another carbon, it can have either a single, double, or triple bond with that carbon. A molecule with single bonds is called an alkane, (picture 1,2 - methane, ethane) and it has bond angles of 109.5 degrees. A double bonded molecule is called an alkene with 120 degree angles (picture 3 – ethylene), and molecule with a triple bond is an alkyne with bond angles about the central carbon of 180 degrees. There are several geometries that these different bonds bring. A tetrahedron is formed by single bonds, trigonal planar from double and linear from triple bonds. This occurs because things that are bonded to the central carbon like to be as far apart as possible. Images from: http://www.nyu.edu/pages/mathmol/library/hydrocarbons/

The Role of Hydrocarbons The main focus of my research has been to determine if it is possible to knot a hydrocarbon molecule. By modifying Heather’s proof given earlier, I have shown that it is impossible to make a stuck unknot from purely single bonded carbon molecules (alkane) This picture is an attempt to create a knot out of single and double bonded carbons (alkene). This molecule’s conformation is not very stable, and it unknots itself in its quest to lower its energy. These structures are tested in a vacuum and there are several methods of computing the energies. Unfortunately, the program that was used can give local minimums for energy instead of absolute minimums, but even with a local min, there is conformation of a semi-stable molecule. Image Created in: HyperChem Professional version 7.5 Image Created in: HyperChem Professional version 7.5

Stuck Hydrocarbon Chain This hydrocarbon, containing single, double, and triple bonds, is a stuck unknot. The length of the ‘tails’ of triple bonded carbons is greater than the rest of the chain, and so it keeps the molecule knotted. The tails are also made of alkynes, which are very rigid and don’t bend. This There are several differences between this molecule and the previous one. This molecule’s “legs” are long strands of triple bonded carbons, with a center that is made of single and double bonded carbons. This molecule is designated an “ene-yne” because of both double and single bonds present. Another possible way to create rigidity in the molecule would be to use allenes, which are double bonded carbons which have double bonds on either side of the carbon, which would result in the long rigid “stick”, but allenes are very reactive, and so would decrease stability, instead of increasing it. Image Created in: HyperChem Professional version 7.5 Image Created in: HyperChem Professional version 7.5

Why? Why not… Chemists like to build interesting molecules. These dancers are secured to a gold plate, and they have no real function except to show that they can be made. There are different molecules that can be used to create different heads, legs, and arms on the molecules. The ones shown are connected, called circle dancers. While these may not look related, if a large enough string of dancers was creates, they could be potentially knotted, or a large game of “Human Knot” could be played. The stuck molecule has a structure that is a “Pi – way” which is a structure that has alternating bonds double-single-double, or triple-single-triple. This gives this molecule conducting properties. This could have interesting results if it was created. INSERT PICTURE OF DANCERS Stephanie H. Chanteau and James M. Tour, J. Org. Chem, Vol. 68, No. 23, 2003. pg. 8759.

Conclusion I have shown that hydrocarbons can make stuck unknots in certain situations. This research has not eliminated the possibility of stuck hydrocarbons with only alkenes or alkanes, but it has proven that stuck unknots can be made by a combination of single, double, and triple bonds.

Thank You Here are the people who helped me greatly in bringing this project together: Dr. Jo Ellis-Monaghan – SMC Dr. Kathleen Mondanaro – SMC Dr. Heather Johnston – Vassar College Dr. Tim Comar – Benedictine University Mary Cox – Grad Student at UVM Whitney Sherman – Undergrad Student at SMC Thank you all for your support.

Sources Almost Regular Knots, Timothy D. Comar and Jessica M. Tyrus, Benedictine University. Stephanie H. Chanteau and James M. Tour, J. Org. Chem, Vol. 68, No. 23, 2003. pg. 8759. Non Trivial Embeddings of Polygonal Intervals and Unknots in 3-Space, Heather Johnston and Jason Cantarella, J.Knot Theory Ramifications 7 (1998), no. 8, 1027-1039. The Knot Book, Colin C. Adams. W. H. Freeman and Company, NY © 1994. www.aperfectworld.org/ metaphors.htm http://pass.maths.org.uk/issue15/features/knots/ http://www.ualberta.ca/CNS/RESEARCH/3DPrinter/ImageGallery/knot.dir.1024x1004.jpg http://www.nyu.edu/pages/mathmol/library/hydrocarbons/ www.knotplot.com

Bridge Number An overpass is shown in the first picture as a strand that goes over one or more other strands. A maximal overpass is where the over strand goes from under crossing to under crossing, the ends ending just before the crossing. The bridge number of a knot is the number of maximal overpasses in that projection. The bridge number of a knot is the minimum number of overpasses in any projection of that knot. Image: Mary

More Stick Knot Equations This equation holds true for b(K) being the bridge number of a knot, alpha being the angle between the sticks between, and not including 0 and Pi. S1,alpha(K) is the regular stick number of K. Equation from: Almost Regular Knots, Timothy D. Comar and Jessica M. Tyrus, Benedictine University.