Depicted here are representatives for the components of the spaces of 5 and 6 move plumbers' knots. The components were enumerated using this 
java code and the images were created using this Mathematica notebook. To classify them, the Jones polynomial for each knot was computed using this 
Mathematica notebook and the Knot Theory` Mathematics package. 

Representatives of components in K5:

Unknot

3_1 (right handed)

3_1 (left handed)

Representatives of components in K6


Unknot

3_1 (right handed)

3_1 (left handed)

4_1

5_1 (right handed)

5_1 (left handed)

5_1 (right handed)

5_2 (left handed)