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Grade 9/10 Math Circles
An Introduction to Group Theory Part 3 - Problem Set

  1. Identify which of the below diagrams are \(3\)-braids.

    A description of the diagram follows.

    Solution: Diagrams 1,2,5, and 6 are all \(3\)-braids. Diagram 3 is not a \(3\)-braid because it has a knot. Diagram 4 is not a \(3\)-braid because not every dot has a string attached to it.

  2. Compute the following two concatenations:

    Two 4-braids with an asterisk between them. In both braids, left dots 1,2,3,4 connect to corresponding right dots 1,2,3,4. In the left braid, the second string starts left 2, goes under then over the third string, ending at right 2. In the right braid, the first string starts at left 1, goes over then under the second string, ending at right 1; also, the fourth string starts at left 4, goes under then over the third string, ending at right 4.

    Two identical braids with an asterisk between them. In the braids, left dot 1 connects to right dot 2 and left dot 2 connects to right dot 1 with a string going over the previous one; also, left dot 3 connects to right dot 4 and left dot 4 connects to right dot 3 with a string going under the previous one.

    Solution: The first concatenation is as follows.

    Left dots 1,2,3,4 connect to corresponding right dots 1,2,3,4. The third is horizontal. The second goes under then over the third near its left end. The fourth goes under then over the third near its right end. The first goes over then under the second near its right end.

    The second concatenation is as follows.

    Left dots 1,2,3,4 connect to corresponding right dots 1,2,3,4. The second string goes over then under the first. The fourth string goes under then over the third.

  3. Compute the inverses of the following two braids:

    A 4-braid with left dots 1,2,3,4 connected to corresponding right dots 1,2,3,4. The second string starts at left 2, goes under the third string connecting 3 to 3 and then over the same string, ending at right 2.   A 4-braid with left dots 1,2,3,4 connected to corresponding right dots 1,2,3,4. The first string starts at left 1, goes over the second string connecting 2 to 2 and then under the same string, ending at right 1. The fourth string starts at left 4, goes under the third string connecting 3 to 3 and then over the same string, ending at right 4.

    Solution: The inverse of the first braid is as follows.

    A 4-braid that matches the first braid except the second string goes over then under the third instead of under then over.

    The inverse of the second braid is as follows.

    A 4-braid that matches the second braid except the first string goes under then over the second string, and the fourth string goes over then under the third string.

  4. Consider the set \(B_n\) of all \(n\)-braids, where \(n\in\{1,2,3,\dots\}\). Is \(B_n\) finite or infinite?

    Solution: The set \(B_n\) is finite when \(n=1\) and infinite when \(n\geq 2\). Let’s see why this is true. First consider the case when \(n=1\). There is only one \(1\)-braid, and it is

    One dot on the left connected by a string to one dot on the right.

    This means that there is only one element in \(B_1\), and so \(B_1\) is finite. Next, consider the case when \(n=2\). To see why \(B_2\) is infinite, consider the following \(2\)-braids:

    Six 2-braids, each with a horizontal string connecting left dot 1 to right dot 1. In the first braid, a horizontal line connects left 2 to right 2. In the second, the string starts at left 2, goes over then under the other string, and ends at right 2. In the third, the string starts at left 2, goes over then under, then goes over then under a second time, and ends at 2. This pattern continues with the second string going over and under three times, then four times, etc.

    In all of these \(2\)-braids, we see that the second string wraps around the first string a various number of times. From left to right and top to bottom, we see that the number of wraps increases. From left to right, the \(2\)-braids in the top row have 0 wraps, 1 wrap, and 2 wraps. From left to right, the \(2\)-braid in the bottom row have 3 wraps, 4 wraps, and 5 wraps. We can keep increasing the number of these wraps forever, and so there are an infinite number of \(2\)-braids.
    Lastly, consider the case when \(n\geq 3\). We will use \(B_2\) to show that there are an infinite number of \(n\)-braids. We construct \(n\)-braids as follows. Take the last \(n-2\) strings and dots to be

    One dot on the left connected by a string to one dot on the right.

    Then, we can think of the the remaining string connections (aka top 2 strings and pairs of dots) as a \(2\)-braid. In other words, we have an "embedding" of \(B_2\) into \(B_n\). By varying the top two string connections through the elements of \(B_2\), we generate an infinite number of \(n\)-braids (since \(B_2\) is infinite). So, \(B_n\) is infinite. Here are some concrete examples of \(3\)-braids constructed in this way:

    Six 3-braids, each horizontal strings connecting dot 1 to dot 1 and dot 3 to dot 3. In the first braid, the second string has 0 wraps. In the second braid, the second string has 1 wrap around the first string. In the third braid, the second string has 2 wraps around the first string. This pattern continues in the next three braids adding more wrap each time.

    We see that the top two string connections are just the \(2\)-braids from above.