Black Swan (黑天鹅 Hei Tian'e) is a thriller film directed by Darren Aronofsky based on Tchaikovsky's Swan Lake.
She does dot, shreds def, dmg bonus buff, and has aoe. This already makes her a very strong character. In Genshin, there are not many characters that can shred def and deal a lot of aoe damage at C0.
Her BA has a 65% chance (lv6) of inflicting Arcana on the enemy. If the enemy is inflicted with a dot (wind shear, shock, burn, bleed), each dot also has 65% chance of inflicting 1 extra Arcana stack. So suppose an enemy has the max possible of 4 dots (not sure how possible this is, would need something like Kafka/Serval, Luka, Guinaifen or a full diversity team). The average number of Arcana stacks from BA would be \(1+4*.65 = 3.6\).
Her E does a blast and has 100% chance of inflicting Arcana on the enemies. It also has 100% chance of shredding their def by around 20% at lv10 (for 3 turns).
So what on earth is Arcana? Arcana basically acts like an enhanced wind dot. Additional stacks on Arcana make it do much more damage (like literal nukes per turn). Moreover, Arcana can spread like a contagion to adjacent targets and even ignore def when there are enough stacks. This spreading property combined with high scaling and def ignore make Arcana pretty broken if we may use a moderate and sober term. This dynamical spreading effect (deterministic) is reminiscent of Arataki Itto's C6 (which is probabilistic/stochastic). Actually, technically the spreading effect may still be a bit random as it could depend on the chance (effect hit rate) that Swan and other characters apply dot.
Arcana can stack up to 50 times. At lv10 it does 240% atk damage to the target. Each stack increases the scaling by 12%. So at 50 stacks, it is 840% atk in a turn, which is broken.
Arcana dmg is dealt at the end of queue, after other dots. The dmg is dealt each turn and resets to 1 stack after dealing dmg each turn.
She gets 10% ehr from traces so the 65% base chance increases to around 70%. One of her passives increases her dmg based on 60% of her ehr, up to 72% (max reached at 120% ehr). So she wants ehr on her relics.
With another passive, when her E hits an enemy, each dot on the enemy has 65% chance of inflicting an additional stack of Arcana. So her E becomes like her BA and on average inflicts 3.6 stacks (at 0 ehr). This passive is almost wasted as it could have been included as part of her E to begin with. Swan can then get some other additional passive.
With another passive, when an enemy enters battle there is 65% chance for it to get 1 Arcana stack. When an ally deals dot damage to it, there is 65% chance for it to get 1 more stack of Arcana, up to 3 times in one attack. So perhaps this passive is for Kafka.
Her Q does initial aoe damage and lasts 2 turns. During this time, the Arcana debuff is treated like a dot (so the above passives will proc on it). Moreover Arcana will not reset during these 2 turns. Also, enemies take 25% more dmg for 2 turns (at lv10).
Swan is already a pretty broken dot dps at E0, so it is a pleasant surprise that her eidolons are also quite good.
E1 shreds res on dot-afflicted enemies (for wind, physical, fire, lightning).
E2: when an enemy is defeated it will inflict 6 Arcana stacks on adjacent targets. This is better for aoe than st.
E4 reduces enemy effect res and gives Swan energy.
E6: when allies attack enemies, Swan has 65% chance of inflicting Arcana on them. Moreover, each time she inflicts Arcana, she has 50% to inflict 1 additional stack. In practice this may make it easier to hit the max of 50 stacks.
Teammates: Serval, Sampo, Luka, Guinaifen, Kafka, healer, Tingyun
By the way, the people translating the Chinese to English for all her trace and eidolon names can do a better job. However, Arcana is a perfect translation.
Now it's time for our favorite section: how would you design Swan's kit in Genshin?
Say she is an anemo catalyst.
When her NA crits it inflicts Arcana on the enemy. The mark can be like say Nahida's E. Every say 3s enemies inflicted with Arcana will take damage. Her E applies Arcana in aoe and shreds def. Her Q makes her debuff last longer so that it can accumulate more stacks. Enemies will also receive 25% more dmg during her Q.
Arcana can accumulate stacks. The more stacks the more damage it does.
A passive can increase her dmg bonus based on her crit rate. Arcana can deal damage when Swan is off field. When allies deal damage they can inflict more stacks of Arcana.