There is a classical approach by Birkhoff and Lewis, which remained dormant for decades. It was recently revived by Cautis and Jackson (start herehere [“The matrix of chromatic joins and the Temperley-Lieb algebra”, J. Combin. Theory 89 (2003), 109–155] and proceed herehere [“On Tutte's chromatic invariant”, Trans. Amer. Math. Soc. 362 (2010), 491–507]), using the Temperley-Lieb algebra.