What is...the Tutte polynomial?

Goal. I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much. This time. What is...the Tutte polynomial? Or: Counting using polynomials. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Slides. http://www.dtubbenhauer.com/youtube.html Tutte polynomial. https://en.wikipedia.org/wiki/Tutte_polynomial https://mathworld.wolfram.com/TuttePolynomial.html https://www.math.ucdavis.edu/~deloera/MISC/LA-BIBLIO/trunk/Welsh/welsh-tutte-polynomial.pdf https://math.berkeley.edu/~brandtm/talks/tutte.pdf https://homepages.dcc.ufmg.br/~gabriel/seminars/coutinho_tuttepolynomial_seminar.pdf Variations. https://en.wikipedia.org/wiki/Graph_polynomial https://en.wikipedia.org/wiki/Knot_polynomial https://arxiv.org/abs/1610.01839 https://web.mat.upc.edu/anna.de.mier/tesidina4.pdf Tutte polynomial and Jones polynomial. https://en.wikipedia.org/wiki/Tutte_polynomial#Jones_polynomial https://www.ams.org/journals/proc/1988-103-02/S0002-9939-1988-0943099-0/S0002-9939-1988-0943099-0.pdf Background. https://en.wikipedia.org/wiki/Graph_theory https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#Undirected_graph https://en.wikipedia.org/wiki/Loop_(graph_theory) https://en.wikipedia.org/wiki/Bridge_(graph_theory) https://en.wikipedia.org/wiki/Deletion%E2%80%93contraction_formula https://en.wikipedia.org/wiki/Tree_(graph_theory) https://en.wikipedia.org/wiki/Eulerian_path#Definition https://en.wikipedia.org/wiki/Robbins%27_theorem Mathematica. https://reference.wolfram.com/language/ref/TuttePolynomial.html Pictures used. https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)#/media/File:6n-graf.svg https://en.wikipedia.org/wiki/Loop_(graph_theory)#/media/File:6n-graph2.svg https://en.wikipedia.org/wiki/Bridge_(graph_theory)#/media/File:Graph_cut_edges.svg https://en.wikipedia.org/wiki/Tutte_polynomial#/media/File:Deletion-contraction.svg https://en.wikipedia.org/wiki/Tree_(graph_theory)#/media/File:Tree_graph.svg https://media.springernature.com/lw785/springer-static/image/chp%3A10.1007%2F978-1-4614-6971-1_5/MediaObjects/301849_1_En_5_Fig9_HTML.gif YouTube and co. https://www.youtube.com/watch?v=4Pd1ohALKqs https://www.youtube.com/watch?v=z2oTSs1iJvc