Knots and quandles
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Knots and quandles book
This chapter aims to determine if the edges of an oriented knot diagram can be labeled with the elements of a quandle. The labeling must satisfy the relations determined by the crossings in the diagram. The chapter utilizes linear algebra to find a solution to the system of equations determined by the knot quandle. Finding these solutions is challenging because not all systems of equations have a non-trivial integer solution. The Alexander polynomial of an oriented virtual knot diagram is computed using the same strategy. Select a point exterior to the knot diagram. Label each overcrossing with a point. Draw a path from the overcrossing point to the exterior point. If the path intersects the diagram an even number of times, the row corresponding to the crossing has coefficient 1.