The Jones representation of genus 2 at the 4th root of unity and the Torelli group

The linear representation of the mapping class group of a closed orientable surface of genus 2 defined by V. Jones is examined. It arises from the Iwahori–Hecke algebra representations of Artinʹs braid group of 6 strings. Under a certain nontrivial specialization, the image of the Torelli group is explicitly computed. As an application, the relation with the Rochlin invariant of homology 3-spheres is discussed.