V. Jones found a concrete way to construct knots and links from elements of Thompson’s group $F$ which is an interesting finite presented infinite group having many counter-intuitive properties where Aiello has summurised the program as Thompson knot theory. Properties of $F$ from the viewpoint of combinatorial group theory were much investigated where the conjugacy problem of $F$ has been solved by Brin and Squier, Guba and Sapir and lately presented by Belk and Matucci from a more dynamical perspective by using the so-called (annular) strand diagrams. In an attempt to tackle the Markov theorem of $F$, we found that there could be an interested relation between conjugacy classes of $F$ and Thompson knot theory by considering annular strand diagrams related to the group elements and we proved that for any link, there exist elements from infinitely many conjugacy classes of Thompson’s group $F$ that realise it via Jones’ construction. This is a joint work with Yuanyuan Bao (arXiv:2504.01714).