Structure and Operations

(For In[1] see Setup)

Thus here's one tautology and one easy example:

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And another easy example:

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For example,

The connected sum [math]\displaystyle{ K=4_1\#4_1 }[/math] of the knot 4_1 with itself has 8 crossings (unsurprisingly):

It is also nice to know that, as expected, the Jones polynomial of [math]\displaystyle{ K }[/math] is the square of the Jones polynomial of 4_1:

It is less nice to know that the Jones polynomial cannot tell [math]\displaystyle{ K }[/math] apart from the knot 8_9:

But [math]\displaystyle{ K=4_1\#4_1 }[/math] isn't equivalent to 8_9; indeed, their Alexander polynomials are different: