Primitive Elements in the Free Group of Rank Two
Primitive Elements in the Free Group of Rank Two
This Demonstration determines whether or not a word in the free group of rank two is primitive. Let and . The word is first cyclically reduced, that is, and so on. If it contains both and or and , then the word is not primitive. Otherwise, the word is written along the unit circle moving between letters, where is the length of the word and , where and are the absolute values of the exponent sums of and , respectively. A red disk indicates and a blue disk indicates . If either of the two letters occupy the same location on the circle, or if the letters are not in two contiguous groups, the element is not primitive.
A=
-1
a
B=
-1
b
axA=x
a
-1
a
b
-1
b
2πk
w
w
k=min{,}
w
a
w
b
w
a
w
b
a
b
±1
a
±1
b