The Alexander polynomial is a link invariant and is used in this thesis to analyze the spatial structure of proteins. It can efficiently be computed by the Jacobianadjugate method, derived from the...Show moreThe Alexander polynomial is a link invariant and is used in this thesis to analyze the spatial structure of proteins. It can efficiently be computed by the Jacobianadjugate method, derived from the adjugate of a Jacobian matrix of the link group. It does so by building a link diagram from smaller tangle diagrams using the operations disjoint union (t) and stitching (mij stitches edge i to edge j) and similarly computing the Jacobian-adjugates JA(T) of the tangle diagrams T. This Jacobian-adjugate is a pair (∆, A) with ∆ ∈ Z[Gab] and A ∈ Mat(Z[Gab]), where G is the group of a link diagram. It satisfies the following rules: (∆A, A) t (∆B, B) = (∆A · ∆B, ∆B · A ⊕ ∆A · B) mij ((∆, A)) = (∆ − A i j ,(1 − ∆−1A i j )A bi + ∆−1A bi j · A i ). Positive and negative crossings with over-strand a and outgoing and incoming under-strand edges b and c have as Jacobian-adjugates (∆ is written in the upper left corner) 1 a b c a 1 0 1 − b b 0 1 a and 1 a b c a 1 0 a −1 (b − 1) b 0 1 a −1 . The Alexander polynomial equals the gcd of the remaining 1 × n matrix when there is just one stitching left. In the single-variable case, at each stage of the computation we only need those columns corresponding to incoming edges, and the final 1 × 1 matrix gives the Alexander polynomial. This Alexander polynomial can be used to compute topological invariants of proteins. Using Sage code we can reconstruct proteins from files from the Protein Data Bank and compute certain (single-variable) Alexander polynomials corresponding to it. Firstly, a link is obtained by closing chains along hydrogen bonds. Running the code on a sample of 1200 PDB files suggests that almost no protein contains a knotted structure. So it seems that the methods of studies such as [2] often infer a knotted structure that is absent in the actual protein. Secondly, a link is obtained from the protein by replacing chains and hydrogen bonds by mutually linked loops. The corresponding Alexander polynomial is very large and some improvements can still be made with regard to the computation time.Show less
This work follows - hopefully somewhat coherently - a small journey through hyperbolic geometry and what I found most interesting in it. It is intended as a spyglass to look at some aspects of the...Show moreThis work follows - hopefully somewhat coherently - a small journey through hyperbolic geometry and what I found most interesting in it. It is intended as a spyglass to look at some aspects of the subject for a student who finds himself interested, even only in knowing what's it about. Hyperbolic geometry is a subject barely mentioned at undergraduate level, and rarely studied in general first- or second-year graduate courses. This comes quite surprisingly since the sheer simplicity of the geometric intuition - one of two possible negations of Euclid's parallel postulate - was first formalized by Gauss with the notion of curvature around 200 years ago. One of the reasons is possibly that the difficulty of the questions grows rapidly, and even at a medium level the study of hyperbolic manifolds borrows tools and techniques from a vast array of subjects: differential and algebraic geometry, complex analysis, representation theory, homological algebra, just to mention the most important ones. On the other hand, this same study helps in dealing with many topological questions not directly connected to it, arising e.g. from the study of knots, and has links to theoretical physics. Due to my background and inclination I've been more attracted to the algebraic - and at times combinatorial - aspects of the theory. This is of course reflected also in the choice of topics. It goes without saying that a master thesis is meant to be as useful to its writer as it is to its readers. My hope is for the present work to prove itself as useful for a reader as it was for me to write it.Show less
