Fractality and Topology of Self-Avoiding Walks

We have analyzed geometric and topological features of self-avoiding walks. We introduce a new kind of walk: the loop-deleted self-avoiding walk (LDSAW) motivated by the interaction of chromatin with the nuclear lamina. Its critical exponent is calculated and found to be different from that of the ordinary SAW. Taking the walks as point-clouds, the LDSAW is a subset of the SAW. We study the difference between the LDSAW and SAW by comparing their fractal dimensions and growth rates of the Betti number. In addition, the spatial distribution of the contacts inside a SAW, which is also a subset of SAW, is analyzed following the same routine. The results show that the contact-cloud has a multi-fractal property and different growth rates for the Betti number. Finally, for comparison, we have analyzed random subsets of the SAW, showing them to have the same fractal dimension as the SAW.