Alphabeta MathGeometric Braids and Artin Generators
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 7 results · all verified · 4 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Category Theory
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Definition
Geometric braids in the disc with setwise endpoints
Definition
Braid isotopy relative to the top and bottom endpoints
Proposition
Stacking of geometric braids is a well-defined associative operation on isotopy classes
Theorem
The isotopy classes of geometric braids based at Q form a group, and the endpoint permutation is a homomorphism
Definition
The elementary geometric half twist, its support disc, and its opposite
Lemma
Far commutativity of elementary geometric half twists
Lemma
The geometric three strand braid relation
Lemma
Geometric braids admit generic polygonal representatives
Lemma
Every geometric braid is isotopic to a stacking of signed elementary half twists
Proposition
The Artin presentation surjects onto the geometric braid group
Sources
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.2-1.3, printed pp. 4-5
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.1, author manuscript pp. 3-5
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.1, author manuscript pp. 3-4
- Juan Gonzalez-Meneses, Basic results on braid groups, section 1.5, printed pp. 7-8, Figure 2
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, section 1.2, author manuscript pp. 5-6
- Juan Gonzalez-Meneses, Basic results on braid groups, section 1.5, printed pp. 7-8
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.5 and 3.2, printed pp. 7-8 and 23-26
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.4-1.5, printed pp. 6-8
- Maurice Chiodo, An Introduction to Braid Theory, section 2 (crossing decomposition), pp. 8-12
- Joan S. Birman and Tara E. Brendle, Braids: A Survey, sections 1.2-1.3, author manuscript pp. 5-7