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 $K=4_{1}\#4_{1}$ of the knot 4_1 with itself has 8 crossings (unsurprisingly):

It is also nice to know that, as expected, the Jones polynomial of $K$ is the square of the Jones polynomial of 4_1:

It is less nice to know that the Jones polynomial cannot tell $K$ apart from the knot 8_9:

But $K=4_{1}\#4_{1}$ isn't equivalent to 8_9; indeed, their Alexander polynomials are different: