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Essential properties of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
For a different proof that any reduced (twist-free) alternating knot diagram represents a non-trivial link, based on a connection between the interlace polynomial and the Alexander polynomial (a knot invariant preceding the Jones polynomial), see Balister, Bollobás, Riordan and Scott [68]. A generalization of the Jones polynomial was introduced by Freyd, Yetter, Hoste, Lickorish, Millett and Ocneanu [503], and related independent work was undertaken by Przytycki and Traczyk [939]. A state space expansion of this homflyptpolynomialhomflypt polynomial (or homflypolynomial) was given by Bollobás, Pebody and Weinreich [157]. A state space expansion of this homflyptpolynomialhomflypt polynomial was given by Bollobás, Pebody and Weinreich [157]. Unfortunately, even the homflypt polynomial does not distinguish all knots; two knots related by mutation are known to have the same homflypt polynomial (see [503]). For more on these topics see, for example, the books [2, 772, 815].
Topological and threading effects in polydisperse ring polymer solutions
Published in Molecular Physics, 2021
Iurii Chubak, Christos N. Likos, Jan Smrek
Metropolis MC simulations of single ring polymers were performed using translational displacements of separate monomers in combination with collective rotations of ring arcs, the so-called crankshaft moves, as described in [18,21]. A series of such collective crankshaft rotations can potentially result in a knotted ring configuration, thereby making the topological state of the ring variable in the course of a whole simulation run. Nevertheless, the ring's topology can be preserved by imposing additional checks for bond crossing after every trial MC move [3,18,21], which naturally slows down the sampling of independent ring configurations and also limits the magnitude of rotations that can be performed during a typical crankshaft move [18,21]. In this work, we have adopted a slightly different strategy: instead of performing bond crossing checks at every single trial MC move, we simply generated a set of independent ring conformations and then determined their topology separately using the KymoKnot software [22] based on the Alexander polynomial. This enabled us to efficiently sample configurations of relatively large rings with contour length up to and also get insight into the knotting probability in the currently employed off-lattice ring model. We define one MC step as a sequence of N trial translational displacements and one crankshaft move with arbitrarily large rotations of arcs that had maximal length N/3. In such case, almost uncorrelated configurations, as evidenced by the autocorrelation function of the ring radius of gyration, defined below in Equation (3), were sampled every 500 MC steps. For every N considered, we generated two independent runs, each having single ring states.