One possibility is that Annabeth has to face some challenges to prove to Percy that she truly wants him back. She could be on a quest to find something important to Percy, like a lost item from his past. This would show her dedication. Also, there might be some new characters in the fanfic who try to stop her from getting back with Percy, adding more drama to the story.
Annabeth could try to be more understanding of Percy's needs. In the fanfiction, she might give him space at first, but then slowly show up in his life in small, positive ways. Like bringing him his favorite food or helping him with a difficult task without being pushy.
The storyline could develop with Annabeth first reflecting on what went wrong in their relationship. She might write in a journal all the things she misses about Percy. Then she could take the bold step of showing up at Percy's new place unannounced. This would shock Percy but also open the door for communication.
There could also be external factors. For example, there might be a prophecy or some sort of danger that keeps Percy away from Annabeth. He might be so focused on dealing with this external threat that he doesn't have time or energy to think about getting back together with her.
In such a fanfic, the plot could be very complex. Maybe Annabeth realizes she made a huge mistake. She might start by trying to approach Percy subtly, like showing up at his favorite places. But Percy would be hurt and angry. He might initially reject her attempts. However, as she keeps trying, she could start to show real remorse and explain the reasons for her cheating, which could be due to some external influence or her own insecurities. And Percy might gradually start to question his own feelings towards her again.
Percy's reaction could be complex. Initially, he'd be hurt and angry. But as time passes, if Annabeth is sincere in her attempts to get him back, he might start to question his own feelings. He could be torn between his love for her and the pain of the betrayal.
She could start by being honest with Percy about her feelings. Just come clean and say why she wants to get back together. For example, she might say she realized she can't live without him. This straightforward approach might help Percy see that she is sincere.
Annabeth is known for her intelligence and strategic thinking. She is a strong and independent character. Percy, on the other hand, is brave and has a strong connection to the water due to his father being Poseidon. In the fanfic, when Annabeth sits in Percy's lap, it might be a way to show their unique relationship which has developed over the course of their many shared experiences in the Percy Jackson universe.
Annabeth could show her determination by going out of her way to be in the same places as Percy. She'll be wherever he is, whether it's at his favorite spots or events he attends. And she'll always be there with a kind word or a helpful gesture, constantly reminding him of their connection. Also, she might face any danger or difficulty that comes her way while trying to get close to him again, showing that she's willing to do whatever it takes.
In the fanfiction world, their personalities might be very complementary. Annabeth is intelligent and strategic, while Percy is brave and impulsive. This yin - yang dynamic could lead to a strong attraction, and the want to kiss could be a natural outgrowth of that attraction.