This puzzle has 11 strange looking creatures that you may recognise as mathematical knots, especially given the puzzle's title. A mathematical knot is like a piece of stretchy string with its ends joined to form a loop. Two knots are considered to be the same if one can be manipulated in space, without cutting the string, so that it looks like the other. The simplest knot is the unknot which is a loop in space that can be manipulated into a circle. This is the only knot that can be drawn on a flat sheet of paper without crossings. All other knots will necessarily have crossings when projected onto paper. For each knot, there are many angles you could look at it from, it could be stretched and squeezed and it could be given extra unnecessary twists in space, so the number of crossings when a particular knot is drawn is not fixed. However, for each knot there will be a way of drawing it with a minimum possible number of crossings.
Each knot creature in the puzzle is drawn with extra twists which can be undone without cutting and so they are clearly not drawn with a minimum number of crossings. This gives a clue on how to start. Counting the number of crossings of the string for each gives 18, 15, 12, 6, 19, 31, 14, 27, 20, 15, 13 crossings which when taken mod 26 and associated with letters using 1=a, 2=b, 3=c,..., 26=z, 27=a, etc, spells out "
Dale Rolfsen wrote a book called "Knots and Links" in which he published a table of all prime knots which have minimum crossing number of 10 or less. He labeled his knots using Alexander-Briggs notation which groups knots according to their minimum crossing number and then arranges them in an arbitrary but fixed order. So, for example, the third knot with minimum crossing number 6 is labelled 6 The word "Rolfsen" in the first message and knowing this has something to do with knots should lead you to associate a label to each of the knot creatures, but before you can identify them, you first need to remove the excess twists to give a simpler picture like the one below. Note that the knot creatures are drawn so that their centres are already in the correct orientation for easy matching with the way they are presented in the Rolfsen Knot Table.
In this picture, reading from left to right and then top to bottom, the Alexander-Briggs labels for the knots are: 7
But what to do with these labels? The key is the second word in the first message, "atom", which reinforces the hint in the title that this puzzle has something to do with chemistry. Furthermore, the word "atom" suggests that it's the chemical elements in particular that are important. In fact, many will have already guessed that there is a connection with an episode in the history of knot theory. In 1867, Lord Kelvin (Sir William Thomson) published a paper titled
The vortex atoms theory did not successfully associate knots with atoms so it won't tell us how to make the connection. However, since each element is naturally associated with a number, its atomic number, and the knot labels have a main number, perhaps the knot labels could be telling us how to choose letters from elements. Subscripts often refer to an index and so a likely decoding rule is that n Decoding all the knots this way...
...gives the puzzle solution of " | |||||||||||||||||||||||||||||||||||||||||||||||||||||||

The answer is: nucleargyre |
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