2022/11/28 by Mahanta, Kuwari
#57K20 #57M60 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2211.15290
Let Sg denote a closed, orientable surface of genus g ≥ 2 and C(Sg) be the associated curve graph. Let d be the path metric on C(Sg) and a0 and a4 be a pair of curves on Sg with d(a0, a4) = 4. In this article, we fix the vertex a0 and apply the Dehn twist about a4, Ta4, to it in an attempt to create pairs of curves at a distance 5 apart. We give a necessary and sufficient topological condition for d(a0, Ta4(a0)) to be 4. We then characterise the pairs of a0 and a4 for which 5 ≤ d(a0, Ta4(a0)) ≤ 6. Lastly, we give an example of a pair of curves on S2 which represent vertices at a distance 5 in C(S2) with intersection number 144.