3.5: Portfolio Problems

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Writeups of the following problems taken from our Class Outlines \(2\) will constitute the Algebra Portfolio for this semester.

\(3.6.1\) Exercises

Exercise \(3.6.1.1\).

Use a presentation matrix for the figure-eight knot \(4_{1}\) to determine its Alexander polynomial. Then, repeat the process using a different \(3\times 3\) minor of the matrix; how does this change your process? Your answer?

Exercise \(3.6.1.2\).

Following the trefoil example from class, determine the full set of elements in the fundamental kei \(\mathcal{K}(4_1)\) of the figure-eight knot, and a table of operations for this kei. (Note that unlike the trefoil, not all elements of this kei will be arcs in the knot diagram.)

Exercise \(3.6.1.3\).

Thinking about the definition of the knot group, make a conjecture and explain your reasoning: The knot group of the unknot is... (Hint: using the simplest diagram of the unknot is probably enough to get a good idea.)

Exercise \(3.6.1.4\).

Determine the (Wirtinger presentation of the) knot group of the figure-eight knot. Then, try to simplify this presentation so that it contains only two generators.

Hint
In the Wirtinger presentation, one relation is always redundant with the others. So, begin by erasing any one relation; then, try to simplify those that remain.