Abstract
This doctoral thesis offers a new approach to the construction of arbitrarily knotted configurations in physical systems. We describe an algorithm that given any link $L$ finds a function $f:mathbb{C}^2tomathbb{C}$, a polynomial in complex variables $u$, $v$ and the conjugate $overline{v}$, whose vanishing set $f^{-1}(0)$ intersects the unit three-sphere $S^3$ in $L$. These functions can often be manipulated to satisfy the physical constraints of the system in question. The explicit construction allows us to make precise statements about properties of these functions, such as the polynomial degree and the number of critical points of $arg f$.Furthermore, we prove that for any link $L$ in an infinite family, namely the closures of squares of homogeneous braids, the polynomials can be altered into polynomials from $mathbb{R}^4$ to $mathbb{R}^2$ with an isolated singularity at the origin and $L$ as the link of that singularity. Links for which such polynomials exist are called real algebraic links and our explicit construction is a step towards their classification.
We also study the crossing numbers of composite knots and relate them to crossing numbers of spatial graphs. The resulting connections are expected to lead to a new approach to the conjecture of the additivity of the crossing number.
| Date of Award | 25 Sept 2018 |
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| Original language | English |
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| Supervisor | Jonathan M Robbins (Supervisor) |