* - for all degrees (HBₖ,HBₖ₊₁,HBₖ₊₂,...)

+2𝑔𝑛 - shift of homology degree coming from:
2 - full twist
𝑔 - number of negative twists in 𝑋
𝑛 - number of free strands

Read the spatial story:
brainec.com/s/oc82wP8HXxMUMTkZ3dlU

Based on van den Berg et al. (2015), Braid Floer Homology.

𝑏𝑟𝑎𝑖𝑑 - set of strands
𝑠𝑡𝑟𝑎𝑛𝑑 - 1-dimensional curve (in 𝑆¹×𝐷²)
𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 - only positive crossings (strand 𝑖 passes over 𝑖+1)
[𝑋 rel 𝑌] - relative braid class (𝑋 - free strands that twist around 𝑌 - fixed strands)
[𝑋₊ rel 𝑌₊] - positive rel. braid class (𝑋₊ -||- that twist *positively* around 𝑌₊)

PS: Recently been very curious about possible conversion between formalized math theorems/proofs in Lean and their spatial human-readable format optimized for reading and comprehension.

..psychology of learning, transition design. Building a scalable instructional medium and practice of writing high-level material in a way that is aligned with our human cognitive experience. Currently pivoting on mathematics

My pleasure to connect, always 🎩
MSc Thesis: hdl.handle.net/10316/96137

See how Mertens’ Second Theorem enters the final stage of the proof of Lemma 4.4, a key step toward the density-one Davis–Lelièvre theorem:

𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞)
brainec.com/s/PCIEHHEEPE...
Based on: Kontorovich, A., & Zhang, X. (2024). On the local-global conj... arXiv:2409.10682

See how can you write it spatially:
brainec.com/s/ZmkrjF2ml2...

See how can you write it spatially:
www.brainec.com/s/UlzYMrr8rO...

See how can you write it spatially:
www.brainec.com/s/2SIKNGSmeF...

See how can you write it spatially: www.brainec.com/s/9jQ7n4IUXU...

Read the spatial story: www.brainec.com/s/h16CK8pj8u...

Used for:
- estimating sums of arithmetic functions
- discrete “integration by parts” for sums
- transforming hard sums into easier integrals

Read the spatial story:
www.brainec.com/s/SXxDYUDZPw...

See how creating a multiplicative indicator function allows us to prove Lemma 4.4, a key step toward the Density-one Davis–Lelièvre theorem:
www.brainec.com/s/PCIEHHEEPE...

𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
𝑓(𝑛) = 1 - if 𝑛 squarefree, 𝑓(𝑛) = 0 otherwise
𝑁𝑜𝑡𝑖𝑐𝑒: multiplying a non-squarefree number by any squarefree number always gives a non-squarefree number.

Read the short spatial story: www.brainec.com/s/HO31YvCkft...