Ghost characters and character varieties of 2-fold branched covers

Abstract

It is known that for any knot K every (meridionally) trace-free SL 2 (C)representation of the knot group G(K) gives an SL 2 (C)-representation of the fundamental group π 1 (Σ 2 K) of the 2-fold branched covering Σ 2 K of the 3-sphere branched along K. In this paper, we show by using a notion called a ghost character of a knot that for the (4, 5)-torus knot T 4,5 the fundamental group π 1 (Σ 2 T 4,5 ) has an SL 2 (C)-representation which cannot be given by any trace-free SL 2 (C)representation of G(T 4,5 ).2010 Mathematics Subject Classification. Primary 57M27; Secondary 57M25. Key words and phrases. character varieties, ghost characters, τ -equivalent representation.1 Such an SL 2 (C)-representation of π 1 (Σ 2 K) is called a τ -equivalent representation. See [10].