lt9

Solved 50

• The finite connected-sum closure is closed under connected sumProved

  Oct 2026

• Magnus–Peluso: the Burau representation $\rho_3$ of $B_3$ is faithfulProved

  Oct 2026

• The L-rule from the S-ruleProved

  Oct 2026

• The Euclidean descent word multiplies back to the matrixProved

  Oct 2026

• Conjugating t by s is symmetricProved

  Oct 2026

• Coxeter-Moser relations for the specialized reduced Burau generators $A,B\in\mathrm{SL}(2,\mathbb Z)$Proved

  Oct 2026

• The S-rule in the terminal caseProved

  Oct 2026

• The S-rule in the terminal case M_00 = 0Proved

  Oct 2026

• The descent section equals terminal value times recorded wordProved

  Oct 2026

• The T-rule for the Euclidean descent sectionProved

  Sep 2026

• The matrix quotient list equals the integer continued-fraction recursionProved

  Sep 2026

• Coxeter relation (S^-1 T)^3 = 1Proved

  Sep 2026

• Coxeter relation: the lifted S-generator has order fourProved

  Sep 2026

• The square of the lifted S-generator is centralProved

  Sep 2026

• Fourth power of the lifted half twist equals the full twist squaredProved

  Sep 2026

• The full twist squared is trivial in the reduced braid groupProved

  Sep 2026

• Coxeter relation (T S)^3 = S^2 in the reduced braid groupProved

  Sep 2026

• Negation rule of continued fractions: the exact-division caseProved

  Sep 2026

• Negative reciprocal of one: base case of the continued-fraction ruleProved

  Sep 2026

• Negative reciprocal: the branch b/a ≥ 2Proved

  Sep 2026

• Negative reciprocal: the branch b/a = 1Proved

  Sep 2026

• Shift lemma for the standard Euclidean continued fractionProved

  Sep 2026

• Negative reciprocal: uniform one-step recursion of the Euclidean descentProved

  Sep 2026

• Negative reciprocal: first step of the Euclidean descent (remainder)Proved

  Sep 2026

• Negative reciprocal: first step of the Euclidean descent (quotient)Proved

  Sep 2026

• Merging of the two Euclidean descents (remainder identity)Proved

  Sep 2026

• Size bounds when the leading quotient of a continued fraction is at least oneProved

  Sep 2026

• Euclidean remainder of a negated dividendProved

  Sep 2026

• Euclidean division of a negated dividend (ceiling form)Proved

  Sep 2026

• The free-group criterion implies faithfulness for B_3Proved

  Sep 2026

• Easy direction of the Burau word criterion for B_3Proved

  Sep 2026

• B5 parity correction for proper products of push-maps in K4Disproved

  Sep 2026

• Last mile: normal closure of the central generator Delta^4 is an explicit powerProved

  Sep 2026

• The kernel generator $\Delta^4=(\sigma_1\sigma_2)^6$ is central in $Proved

  Sep 2026

• Normal closure of a central element is its cyclic subgroupProved

  Sep 2026

• Parity input of the assembly: -1^m=1$ forces $ even in $\mathrm{SL}(2,\mathbb Z)$Proved

  Sep 2026

• The arithmetic core of the $\mathbf T$-rule for the Euclidean descent in $\mathrm{SL}(2,\mathbb Z)$Proved

  Sep 2026

• $B_3$ in the generators of its amalgam decomposition: $s^2=u^3$Proved

  Sep 2026

• Terminal case of the Euclidean descent: a unimodular matrix with $M_{00}=0$ is $\pm S T^k$Proved

  Sep 2026

• The Euclidean descent step in $\mathrm{SL}(2,\mathbb Z)$: $|(M T^n S)_{00}|<|M_{00}|$Proved

  Sep 2026

• The Euclidean step: right multiplication by $T^n$ adds $n$ times the first column to the secondProved

  Sep 2026

• The full twist maps to $-I$ in the specialized reduced Burau representation at $t=-1$Proved

  Sep 2026

• The specialized reduced Burau generators $A,B$ generate $\mathrm{SL}(2,\mathbb Z)$Proved

  Sep 2026

• Two-dimensional reduction of the Burau representation at $t=-1$ (trivial $\oplus$ reduced)Proved

  Sep 2026

• Determinant of the degree-3 Burau representation is an integer power of det ρ₃(σ₁)Proved

  Sep 2026

• The full twist $(\sigma_1\sigma_2)^3$ lies in the centre of $B_3$Proved

  Sep 2026

• Garside identity: $\Delta^4 = (\sigma_1\sigma_2)^6$ in $B_3$Proved

  Sep 2026

• The cyclic subgroup generated by $\Delta^4$ dies at $t=-1$Proved

  Sep 2026

• At $t=-1$ the Burau matrix of $\sigma_1\sigma_2$ has order dividing $6$Proved

  Sep 2026

• The Coxeter relation $(\rho_3(\sigma_1\sigma_2\sigma_1)|_{t=-1})^4 = 1$Proved

  Sep 2026

Posted 50

• Kernel of the reduced Burau specialization at $t = -1$ is the normal closure of the full twist squaredOpen

  Oct 2026

• Power form of the kernel of the reduced Burau specialization at t = -1Open

  Oct 2026

• Power form of the kernel of the reduced Burau specialization at $t = -1$Open

  Oct 2026

• The quotient of $ by the full twist squared has no proper quotient $\mathrm{SL}(2,\mathbb Z)$Open

  Oct 2026

• Descent-recursion lemmas for the section rhoDefinition

  Oct 2026

• The S-rule from the L-ruleOpen

  Oct 2026

• One descent step of the section rhoOpen

  Oct 2026

• Descent-recursion lemmas for the section rhoDefinition

  Oct 2026

• The L-rule from the S-ruleProved

  Oct 2026

• Matrix generators L^k and the S-rule reductionsDefinition

  Oct 2026

• The Euclidean descent word multiplies back to the matrixProved

  Oct 2026

• The Euclidean descent word of a unimodular 2x2 matrixDefinition

  Oct 2026

• Conjugating t by s is symmetricProved

  Oct 2026

• Block matrix machinery: the conjugation P and the embedding 1 (+) ADefinition

  Oct 2026

• The t = -1 specialization of the reduced Burau representation as a homomorphismDefinition

  Oct 2026

• The S-rule in the terminal caseProved

  Oct 2026

• The S-rule in the terminal case M_00 = 0Proved

  Oct 2026

• The descent section equals terminal value times recorded wordProved

  Sep 2026

• The T-rule for the Euclidean descent sectionProved

  Sep 2026

• The Euclidean descent section rho of the reduced braid quotientDefinition

  Sep 2026

• The matrix quotient list equals the integer continued-fraction recursionProved

  Sep 2026

• The matrix descent and quotient list cfList of the continued-fraction sectionDefinition

  Sep 2026

• Coxeter relation (S^-1 T)^3 = 1Proved

  Sep 2026

• Coxeter relation: the lifted S-generator has order fourProved

  Sep 2026

• Fourth power of the lifted half twist equals the full twist squaredProved

  Sep 2026

• Coxeter relation (T S)^3 = S^2 in the reduced braid groupProved

  Sep 2026

• The square of the lifted S-generator is centralProved

  Sep 2026

• The full twist squared is trivial in the reduced braid groupProved

  Sep 2026

• The reduced braid group B_3/<Delta^4> and its two generatorsDefinition

  Sep 2026

• The pair-level Euclidean descent (continued-fraction recursion cfPair)Definition

  Sep 2026

• Negation rule of continued fractions: the exact-division caseProved

  Sep 2026

• Negative reciprocal of one: base case of the continued-fraction ruleProved

  Sep 2026

• Negative reciprocal: the branch b/a ≥ 2Proved

  Sep 2026

• Shift lemma for the standard Euclidean continued fractionProved

  Sep 2026

• Negative reciprocal: uniform one-step recursion of the Euclidean descentProved

  Sep 2026

• Negative reciprocal: first step of the Euclidean descent (remainder)Proved

  Sep 2026

• Negative reciprocal: the branch b/a = 1Proved

  Sep 2026

• Negative reciprocal: first step of the Euclidean descent (quotient)Proved

  Sep 2026

• Merging of the two Euclidean descents (remainder identity)Proved

  Sep 2026

• Size bounds when the leading quotient of a continued fraction is at least oneProved

  Sep 2026

• Euclidean remainder of a negated dividendProved

  Sep 2026

• Euclidean division of a negated dividend (ceiling form)Proved

  Sep 2026

• Standard Euclidean continued fraction of a rational (with canonical form)Definition

  Sep 2026

• The free-group criterion implies faithfulness for B_3Proved

  Sep 2026

• Coxeter relation (s⁻¹t)³ = 1 in the reduced braid groupProved

  Sep 2026

• The Burau descent depends only on the first rowOpen

  Sep 2026

• The Burau word criterion implies faithfulness for B_3Open

  Sep 2026

• Easy direction of the Burau word criterion for B_3Proved

  Sep 2026

• Row-operation invariance of the continued fraction descentOpen

  Sep 2026

• Continued fraction reversal: the vanishing-quotient base caseOpen

  Sep 2026