Does every pairing of odd and even number in $[n]$ correspond to some alternating knot? I am reading about the Dowker notation in "The Knot Book" By Collins Adam.
There's a link to the book below 
http://people.math.harvard.edu/~ctm/home/text/books/adams/knot_book/knot_book.pdf
An example of recovering an alternating knot using a sequence of even number
8, 10, 12, 2, 14, 6, 4 
is demonstrated on page 36 to 37. 
The process is you basically draw out the knots starting with the first pairing (1,8), and you circle around when you reach a number that is in one of the existing pairing (See Link above, page 37).
However, in this example, if one swap 8 with 10, then there's no way to get out of the closed loop enclosed by the crossing (2,7) without creating another crossing.
More generally, how do we know the recovery method mentioned for Dowker's notation always work? (That is, given any pairing of odd and even number, how do we know if there's always an alternating knot corresponding to it?)
 A: Not every sequence corresponds to a knot.  This is related to the Gauss word realizability problem.  Given a immersed loop in the plane in general position (that is, all self-intersections are transverse double points) with labeled double points, by following the curve one obtains a list of labels, each label appearing twice -- such a list is known as a Gauss word.  He wondered: is there a simple procedure to determine whether a given Gauss word came from some immersed loop in general position?  (This is a problem Gauss was unable to solve, though he enumerated solutions up to five double points.  Dehn gave an algorithm in 1930s, which Rosensthiel and Tarjan in the 1980s showed, using a nice data structure, could be turned into a linear-time algorithm.)
In the end, the problem is essentially the following.  Gauss words can be thought of as describing a 4-regular graph, and the question is whether that 4-regular graph is planar, subject to constraints at each vertex that the incident edges follow a particular clockwise or counterclockwise order.  It turns out some of the graphs are nonplanar.
I've written up some notes on different representations of knots (section 3 gets to DT codes):
https://math.berkeley.edu/~kmill/2019_8_25/gauss_dt_codes.html
