The Legendrian Knot Visualizer takes generating families of legendrian knots and plots their embeddings in R3. A generating family \(f_x\) for a Legendrian knot is a family of functions whose critical values encode a projection of the knot. It turns out there's a diffeomorphism between the critical locus of f and the embedding \[\{(x,y,z) | \frac{\partial f}{\partial e} = y \text{ and } f(x,e)=z \}\]

Text Entry Rules: Polynomials must be written using Python math operators. Meaning ^ must be replaced with ** for exponentiation (standard +, -, *, /) and careful use of parenthesis to preserve order of operations. Especially among dividends and divisors. The sqrt() function is not yet supported so use **(1/2) instead. When typing your generating family, use e1,e2,... for E variables and a1,a2,... for user determined constants.

Other Notes: The Legendrian knot visualizer will attempt to plot anything, even if it's not Legendrian. Exponential functions are supported, but trig functions may not have all solutions returned/graphed. Multiple sets of graphs created for each a-value will multiply calculation times. If a plot has no points or the page never loads, check if the critical locus of f is empty.