We use Fox calculus to assign a marked polytope to a ‘nice’ group presentation with two generators and one relator. Relating the marked vertices to Novikov–Sikorav homology we show that they determine the Bieri–Neumann–Strebel invariant of the group. Furthermore we show that in many cases the marked polytope is an invariant of the underlying group and that in those cases the marked polytope also determines the minimal complexity of all the associated HNN-splittings. Dedicated to the memory of Tim Cochran 1. Summary of results In this paper, a (2, 1)–presentation is a group presentation π = 〈x, y | r〉 with two generators and one relator. A (2, 1)–presentation π naturally gives rise to a group, which we denote Gπ. We say that a (2, 1)–presentation π is nice if it satisfies the following conditions: (1) r is a non-empty, cyclically reduced word, and (2) b1(Gπ) = 2. To a nice (2, 1)–presentation π = 〈x, y | r〉 we will associate a marked polytope Mπ in H1(Gπ;R). A marked polytope is a polytope together with a (possibly empty) set of marked vertices. Now we give an informal outline of the definition of Mπ (see also Figure 1), a formal definition is given in Section 2.3. Identify H1(Gπ;Z) with Z 2 such that x corresponds to (1, 0) and y corresponds to (0, 1). Then the relator r determines a discrete walk on the integer lattice in H1(Gπ;R), and the marked polytope Mπ is obtained from the convex hull of the trace of this walk: (1) Start at the origin and walk across Z reading the word r from the left. (2) Take the convex hull C of the set of all lattice points reached by the walk. (3) Mark precisely those vertices of C that the walk passes through exactly once. (4) Consider the unit squares that are completely contained in C and which touch a vertex of C. The set of vertices ofMπ is defined as the set of midpoints of all of these squares, and a vertex of Mπ is marked precisely when all the corresponding vertices of C are marked. Date: January 14, 2015. 2000 Mathematics Subject Classification. Primary 20J05; Secondary 20F65, 22E40, 57R19 .