Low complexity algorithms in knot theory

We show that the genus problem for alternating knots with $n$ crossings has linear time complexity and is in Logspace$(n)$. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We show that the genus problem for these knots belongs to $TC^0$ circuit complexity class. We also show, that the equivalence problem for such knots with $n$ crossings has time complexity $n\log (n)$ and is in Logspace$(n)$ and $TC^{0}$ complexity classes.