Average size of random polygons with fixed knot topology

We have evaluated by numerical simulation the average size $R_K$ of random polygons of fixed knot topology $K=\mathrm{\varnothing },3_1,3_1\mathrm{♯}{4}_1$, and we have confirmed the scaling law $R_K^2\sim N^{2{\nu }_K}$ for the number $N$ of polygonal nodes in a wide range; $N=100$ – 2200. The best fit gives $2 \nu_K \simeq 1.11$--1.16withgoodfittingcurvesinthewholerangeof$N$. The estimate of $2{\nu }_K$ is consistent with the exponent of self-avoiding polygons. In a limited range of $N$ ($N\gtrsim 600$), however, we have another fit with $2{\nu }_K\simeq 1.01$ – 1.07, which is close to the exponent of random polygons.