Jones, V. F. R. Ten problems. (English) Zbl 0969.57001 Arnold, V. (ed.) et al., Mathematics: Frontiers and perspectives. Providence, RI: American Mathematical Society (AMS). 79-91 (2000). Firstly, ten problems from the areas of mathematics in which the author has worked have been selected and listed. They are: 1. Find a nontrivial knot \(K\) with \(V_K = 1\). 2. Characterise those elements of \(\mathbb Z[t, t^{-1}]\) of the form \(V_K(t).\) 3. Is the representation of the braid group inside the Temperley-Lieb algebra faithful? 4. Do Vassiliev invariants separate knots? 5. Calculate the set of indices of irreducible subfactors of the hyperfinite \({II}_1\) factor. 6. What functions arise as the PoincarĂ© series of a subfactor? In problems 7, 8 and 9, if \((\cdot)\) is a discrete group \(U(\cdot)\) denote its von Neumann algebra. 7. Is \(U(F_n) \cong U(F_m)\) for \(m \neq n (\geq 2)\)? 8. If \((\cdot)\) is a rigid discrete group and \(\alpha\) an automorphism of \(U(\cdot)\) is there a \(\beta\ni \operatorname{Aut}(\cdot)\) with \(\alpha=\beta\) modulo inner automorphism? 9. Calculate the fundamental group of \(U(SL(3,\mathbb Z)).\) 10. Solve the 2-dimensional square lattice Potts model. Secondly, the problems are combined in four blocks: 1) Knot theoretic problems 1, 2, 3, 4 ; 2) Subfactor problems 5 and 6; 3) Problems 7, 8, 9. General \({II}_1\) factors; 4) Problem 10: statistical mechanics, and, finally, they are discussed in detail.There are given 57 references.For the entire collection see [Zbl 1047.00015]. Reviewer: Sergei Georgievich Zhuravlev (Moskva) Cited in 3 ReviewsCited in 20 Documents MSC: 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57M25 Knots and links in the \(3\)-sphere (MSC2010) Keywords:topology; invariants; knots × Cite Format Result Cite Review PDF