Abstract: The dynamic behaviours of the translocations of closed circular polymers and closed knotted polymers through a nanopore, under the driving of an applied field, are studied by three-dimensional Langevin dynamics simulations. The power-law scaling of the translocation time $\tau$  with the chain length N and the distribution of translocation time are investigated separately. For closed circular polymers, a crossover scaling of translocation time with chain length is found to be $\tau \sim N^{\alpha}$, with the exponent α varying from $\alpha =0.71$ for relatively short chains to $\alpha =1.29$ for longer chains under driving force $F=5$. The scaling behaviour for longer chains is in good agreement with experimental results, in which the exponent $\alpha =1.27$ for the translocation of double-strand DNA. The distribution of translocation time $D(\tau)$ is close to a Gaussian function for duration time $\tau <\tau _{\rm p}$ and follows a falling exponential function for duration time $\tau >\tau _{\rm p}$. For closed knotted polymers, the scaling exponent α is 1.27 for small field force ($F=5$) and 1.38 for large field force ($F=10$). The distribution of translocation time $D(\tau)$ remarkably features two peaks appearing in the case of large driving force. The interesting result of multiple peaks can conduce to the understanding of the influence of the number of strands of polymers in the pore at the same time on translocation dynamic process and scaling property.

Key words: Langevin dynamics, closed polymer, scaling behaviour, distribution of translocation time