On the signature of positive braids and adjacency for torus knots

The objects of study in this thesis are knots. More precisely, positive braid knots, which include algebraic knots and torus knots. In the first part of this thesis, we compare two classical knot invariants - the genus g and the signature σ - for positive braid knots. Our main result on positive braid knots establishes a linear lower bound for the signature in terms of the genus. In the second part of the thesis, a positive braid approach is applied to the study of the local behavior of polynomial functions from the complex affine plane to the complex numbers. After endowing polynomial function germs with a suitable topology, the adjacency problem arises: for a fixed germ f, what classes of germs g can be found arbitrarily close to f? We introduce two purely topological notions of adjacency for knots and discuss connections to algebraic notions of adjacency and the adjacency problem.

Material properties of common suture materials in orthopaedic surgery

Suture materials in orthopaedic surgery are used for closure of wounds, repair of fascia, muscles, tendons, ligaments, joint capsules, and cerclage or tension band of certain fractures. The purpose of this study was to compare the biomechanical properties of eleven commonly used sutures in orthopaedic surgery. Three types of braided non-absorbable and one type of braided absorbable suture material with different calibers (n=77) underwent biomechanical testing for maximum load to failure, strain, and stiffness. All samples were tied by one surgeon with a single SMC (Seoul Medical Center) knot and three square knots. The maximum load to failure and strain were highest for #5 FiberWire and lowest for #0 Ethibond Excel (p<0.001). The stiffness was highest for #5 FiberWire and lowest for #2-0 Vicryl (p<0.001). In all samples, the failure of the suture material occurred at the knot There was no slippage of the knot in any of the samples tested. This data will assist the orthopaedic surgeon in selection and application of appropriate suture materials and calibers to specific tasks.

Positive braids of maximal signature

We characterise positive braid links with positive Seifert form via a finite number of forbidden minors. From this we deduce a one-to-one correspondence between prime positive braid links with positive Seifert form and simply laced Dynkin diagrams, as well as a simple classification of alternating positive braid knots.