aK forgets both knot orientation and mirroring. Its oriented members are the distinct normalized forms of K, rK, mK, and mrK. Its unoriented members are the corresponding u objects.
For example, a3_1 has unoriented members {u3_1, um3_1} and oriented members {3_1, m3_1}. Both collections are sets. The oriented set has at least as many members as the unoriented set.
The name uses the smallest oriented member in numeric knot order. For connected sums, the operator applies to the entire expression, as in a(3_1 # 3_1); mirroring individual summands independently is not allowed.
Positive amphichirality identifies K with mK; negative amphichirality identifies K with mrK; reversibility identifies K with rK. Fully amphichiral means both signs of amphichirality. Chiral means neither sign occurs.
Symmetry columns use recorded evidence and possible stabilizers. Unknown means the available evidence does not decide that property. Distinct displayed members may still be possible duplicates when symmetry classification is incomplete.