A finite presentation of knotted trivalent graphs

While knot theory has been studied since the 19th century (and arguably for thousands of years prior to Gauss), knotted trivalent graphs are objects of relatively recent interest. We will extend the methods used by Thurston to find generators of KTGs in 2002, and use them to determine a finitely generated list of relations, thereby acquiring a finite presentation of knotted trivalent graphs. We will first define a set of knotted trivalent graph diagrams, and reiterate Thurston's result that they are generated by the diagrams for the tetrahedron and the twisted tetrahedron. We will then use a version of the same algorithm to establish that the relations on knotted trivalent graph diagrams are finitely generated by the four relations corresponding to the operations: disjoint union with a tetrahedron, disjoint union with a twisted tetrahedron, connect sum, and unzip. Finally we will extend these results to knotted trivalent graphs themselves by equipping KTGs with a thickening of each edge and considering an extended list of Reidemeister moves for knotted trivalent graphs. By adding generators to account for twisting of thickened edges and relations corresponding to each of these five Reidemeister moves we will complete the finite presentatio