Non-left-orderable surgeries of twisted torus knots
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Date
2017
Authors
Dovhyi, Serhii
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Abstract
The topic of study of this thesis belongs both to knot theory and to group theory. A knot is a smooth embedding of a circle in $\mathbb{R}^3$ or $S^3=\mathbb{R}^3\cup\{+\infty\}$. With any knot $K$ we can do an operation which depends on two integer coefficients $p$ and $q$, called $\frac{p}{q}$ Dehn surgery, resulting in a 3-manifold $M$ denoted by $M:=S^3(K,\frac{p}{q})$. A group is left-orderable if it can be given a total strict ordering which is invariant under multiplication from the left. It is hard to understand Dehn surgery geometrically, but algebraically it is clear - the fundamental group $\pi_1(M)$ equals to the fundamental group of the knot complement of $K$ with one relation added. Although the fundamental group of the knot complement is always left-orderable, $\pi_1(M)$ may not be left-orderable. We study left-orderability of $\pi_1(M)$ in case where $K$ is a twisted torus knot.
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Keywords
Knot theory, Dehn surgery, Orderable group, 3-manifold