Topological constraints can affect both equilibrium and dynamic properties of polymer systems, and can play a role in the organization of chromosomes. Despite many theoretical studies, the effects of topological constraints on the equilibrium state of a single compact polymer have not been systematically studied. Here we use simulations to address this longstanding problem. We find that sufficiently long unknotted polymers differ from knotted ones in the spatial and topological states of their subchains. The unknotted globule has subchains that are mostly unknotted and form asymptotically compact RG(s) ∼ s 1/3 crumples. However, crumples display high fractal dimension of the surface d b = 2.8, forming excessive contacts and interpenetrating each other. We conclude that this topologically constrained equilibrium state resembles a conjectured crumpled globule [Grosberg et al., Journal de Physique, 1988, 49, 2095, but differs from its idealized hierarchy of self-similar, isolated and compact crumples. INTRODUCTIONTopological constraints, i.e. the inability of chains to pass through each other, have significant effects on both equilibrium and dynamic properties of polymer systems [1][2][3] and can play important roles in the organization of chromosomes [3][4][5][6]. Previous theoretical studies suggested that topological constraints per se compress polymer rings or polymer subchains by topological obstacles imposed by surrounding subchains [7][8][9]. This compression makes a subchain of length s form a spacefilling configuration that has an average radius of gyration R G (s) ∼ s 1/3 . Recent simulations of topologically constrained unconcatenated polymer rings in a melt [10][11][12][13][14] have demonstrated the effect of compression into space-filling configurations and confirmed s 1/3 scaling, thus providing strong support to previous conjectures.The role of topological constraints in the equilibrium state of a single compact and unknotted polymer remains unknown. Previous studies [3,7,8] have put forward a concept of the crumpled globule as the equilibrium state of a compact and unknotted polymer. In the crumpled globule, the subchains were suggested to be space-filling and unknotted. This conjecture remained untested for the quarter of the century. Here, we test this conjecture by comparing equilibrium compact states of a topologically constrained and unknotted polymer, referred to below as the unknotted globule, with those a topologically relaxed one, referred to below as the knotted globule (Fig. 1).Recent computational studies examined the role of topological constraints in the non-equilibrium (or quasiequilibrium) polymer states that emerge upon polymer collapse [15][16][17][18][19]. This non-equilibrium state, often referred to as the fractal globule [6,15], can indeed possess some properties of the conjectured equilibrium crumpled globule. The properties of the fractal globule, its stability [20], and its connection to the equilibrium state are yet to be understood.Elucidating the role of topological con...