People will search for you by name on LinkedIn, so be sure to accurately display the common name that you use. There are also some key do's and don't about the name field that all LinkedIn users need to know. Consider leveraging two name options: Gender pronouns and audio name pronunciation.
Your LinkedIn profile can appear in Google and other searches. You have the power to decide exactly what will appear outside of LinkedIn. You can also customize your URL to be more findable and to appear more savvy as a user.