Alphabeta MathGeometric Braids and Artin Generators — Examples
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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.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
✓ 4 results · all verified · 3 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 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Braid Groups
Category Theory
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Covering Spaces and Lifting
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Subspaces, Products, and Quotients
- The Fundamental Group of the Circle
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Example
Geometric two strand braids are integer twists
Example
The three strand geometric braid relation
Counterexample
Setwise endpoints do not make a braid pure
Counterexample
An arbitrary isotopy of arcs need not be a braid isotopy
Sources
- Juan Gonzalez-Meneses, Basic results on braid groups, sections 1.2-1.3, printed pp. 4-6
- 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, sections 1.5 and 3.2, printed pp. 7-8 and 23-26
- 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