Detecting the Linkage in an n -Component Brunnian Link IMUS - PowerPoint PPT Presentation

Conjecture An extension of ∆ to an A ∞ -coalgebra structure on C ( BR n ) induces an A ∞ -coalgebra structure on H ( BR n ) with A primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) A non-trivial n-ary operation ∆ n : H ( BR n ) → H ( BR n ) ⊗ n Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 17 / 33

Conjecture An extension of ∆ to an A ∞ -coalgebra structure on C ( BR n ) induces an A ∞ -coalgebra structure on H ( BR n ) with A primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) A non-trivial n-ary operation ∆ n : H ( BR n ) → H ( BR n ) ⊗ n Trivial k-ary operations for all k � = 2 , n ∆ k : H ( BR n ) → H ( BR n ) ⊗ k Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 17 / 33

Conjecture An extension of ∆ to an A ∞ -coalgebra structure on C ( BR n ) induces an A ∞ -coalgebra structure on H ( BR n ) with A primitive diagonal ∆ 2 : H ( BR n ) → H ( BR n ) ⊗ H ( BR n ) A non-trivial n-ary operation ∆ n : H ( BR n ) → H ( BR n ) ⊗ n Trivial k-ary operations for all k � = 2 , n ∆ k : H ( BR n ) → H ( BR n ) ⊗ k Detects the linkage in a n-component Brunnian link Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 17 / 33

Minnich’s A-infinity Coalgebra Structure on a Polygon Theorem (Minnich, 2017) Let G be an n-gon with initial vertex v 1 , terminal vertex v t , and edges e 1 , e 1 , . . . , e n directed from v 1 to v t . Let ∆ 2 denote the Kravatz diagonal. For k > 2 define ∑ ∑ ∆ k ( G ) = e i 1 ⊗ · · · ⊗ e i k + e i 1 ⊗ · · · ⊗ e i k . 0 < i 1 < ··· < i k < t n ≥ i 1 > ··· > i k ≥ t Then ( C ( G ) , ∂ , ∆ � 2 , ∆ � 3 , . . . ) is an A ∞ -coalgebra. Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 18 / 33

Differential Graded Vector Spaces Let V , W be a graded Z 2 -vector spaces Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 19 / 33

Differential Graded Vector Spaces Let V , W be a graded Z 2 -vector spaces A linear map f : V → W has degree p if f : V i → W i + p Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 19 / 33

Differential Graded Vector Spaces Let V , W be a graded Z 2 -vector spaces A linear map f : V → W has degree p if f : V i → W i + p A differential on V is a linear map ∂ : V → V of degree − 1 such that ∂ ◦ ∂ = 0 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 19 / 33

Differential Graded Vector Spaces Let V , W be a graded Z 2 -vector spaces A linear map f : V → W has degree p if f : V i → W i + p A differential on V is a linear map ∂ : V → V of degree − 1 such that ∂ ◦ ∂ = 0 ( V , ∂ ) is a differential graded vector space (d.g.v.s.) Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 19 / 33

Chain Maps Hom p ( V , W ) denotes the v.s. of degree p linear maps Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 20 / 33

Chain Maps Hom p ( V , W ) denotes the v.s. of degree p linear maps Hom ∗ ( V , W ) is a d.g.v.s with differential δ ( f ) = ∂ W ◦ f + f ◦ ∂ V Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 20 / 33

Chain Maps Hom p ( V , W ) denotes the v.s. of degree p linear maps Hom ∗ ( V , W ) is a d.g.v.s with differential δ ( f ) = ∂ W ◦ f + f ◦ ∂ V f is a chain map iff δ ( f ) = 0 , i.e., ∂ W ◦ f = f ◦ ∂ V Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 20 / 33

Chain Maps Hom p ( V , W ) denotes the v.s. of degree p linear maps Hom ∗ ( V , W ) is a d.g.v.s with differential δ ( f ) = ∂ W ◦ f + f ◦ ∂ V f is a chain map iff δ ( f ) = 0 , i.e., ∂ W ◦ f = f ◦ ∂ V The chain maps in ( Hom ∗ ( V , W ) , δ ) form the subspace of δ -cycles Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 20 / 33

Chain Maps Hom p ( V , W ) denotes the v.s. of degree p linear maps Hom ∗ ( V , W ) is a d.g.v.s with differential δ ( f ) = ∂ W ◦ f + f ◦ ∂ V f is a chain map iff δ ( f ) = 0 , i.e., ∂ W ◦ f = f ◦ ∂ V The chain maps in ( Hom ∗ ( V , W ) , δ ) form the subspace of δ -cycles H ( Hom ∗ ( V , W )) = ker δ / Im δ Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 20 / 33

Chain Homotopies Let f , g : V → W be maps of degree p Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 21 / 33

Chain Homotopies Let f , g : V → W be maps of degree p A chain homotopy from f to g is a map T : V → W of degree p + 1 such that ∂ W T + T ∂ V = f + g Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 21 / 33

Chain Homotopies Let f , g : V → W be maps of degree p A chain homotopy from f to g is a map T : V → W of degree p + 1 such that ∂ W T + T ∂ V = f + g When p = 0 we have ∂ V · · · ← − V i − 1 ← − V i ← − · · · ↓ f + g T � � T · · · ← − W i ← − W i + 1 ← − · · · ∂ W Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 21 / 33

Chain Homotopies Let f , g : V → W be maps of degree p A chain homotopy from f to g is a map T : V → W of degree p + 1 such that ∂ W T + T ∂ V = f + g When p = 0 we have ∂ V · · · ← − V i − 1 ← − V i ← − · · · ↓ f + g T � � T · · · ← − W i ← − W i + 1 ← − · · · ∂ W f + g is a boundary if δ ( T ) = f + g for some chain homotopy T Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 21 / 33

Stasheff’s Associahedra The associahedron K n is an ( n − 2 ) -dimensional polytope that controls homotopy (co)associativity in n variables Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 22 / 33

Stasheff’s Associahedra The associahedron K n is an ( n − 2 ) -dimensional polytope that controls homotopy (co)associativity in n variables Associahedra organize the structural data in the definition of an A ∞ -(co)algebra Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 22 / 33

Stasheff’s Associahedra The associahedron K n is an ( n − 2 ) -dimensional polytope that controls homotopy (co)associativity in n variables Associahedra organize the structural data in the definition of an A ∞ -(co)algebra For each n ≥ 2 , let θ n denote the ( n − 2 ) -dimensional cell of K n Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 22 / 33

A-infinity Coalgebras Defined Let ( V , ∂ ) be a d.g.v.s. For each n ≥ 2 , choose a map α n of deg 0 : α n Hom ∗ ( V , V ⊗ n ) C ∗ ( K n ) − → ∂ ↓ ↓ δ Hom ∗− 1 ( V , V ⊗ n ) C ∗− 1 ( K n ) − → α n and define ∆ n : = α n ( θ n ) Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 23 / 33

A-infinity Coalgebras Defined Let ( V , ∂ ) be a d.g.v.s. For each n ≥ 2 , choose a map α n of deg 0 : α n Hom ∗ ( V , V ⊗ n ) C ∗ ( K n ) − → ∂ ↓ ↓ δ Hom ∗− 1 ( V , V ⊗ n ) C ∗− 1 ( K n ) − → α n and define ∆ n : = α n ( θ n ) ( V , ∂ , ∆ 2 , ∆ 3 , . . . ) is an A ∞ - coalgebra if each α n is a chain map, i.e., δα n = α n ∂ Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 23 / 33

A-infinity Coalgebras Defined Let ( V , ∂ ) be a d.g.v.s. For each n ≥ 2 , choose a map α n of deg 0 : α n Hom ∗ ( V , V ⊗ n ) C ∗ ( K n ) − → ∂ ↓ ↓ δ Hom ∗− 1 ( V , V ⊗ n ) C ∗− 1 ( K n ) − → α n and define ∆ n : = α n ( θ n ) ( V , ∂ , ∆ 2 , ∆ 3 , . . . ) is an A ∞ - coalgebra if each α n is a chain map, i.e., δα n = α n ∂ Evaluating at θ n produces the classical structure relations n − 2 n − i − 1 ( − 1 ) i ( n + j + 1 ) � 1 ⊗ j ⊗ ∆ i + 1 ⊗ 1 ⊗ n − i − j − 1 � ∑ ∑ δ ( ∆ n ) = ∆ n − i i = 1 j = 0 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 23 / 33

Structure Relations ∆ n is a chain homotopy among the quadratic compositions encoded by the codim 1 cells of K n δ ( ∆ 4 ) = ( ∂ ⊗ 1 ⊗ 1 + 1 ⊗ ∂ ⊗ 1 + 1 ⊗ 1 ⊗ ∂ ) ∆ 4 + ∆ 4 ∂ = ( ∆ 2 ⊗ 1 ⊗ 1 + 1 ⊗ ∆ 2 ⊗ 1 + 1 ⊗ 1 ⊗ ∆ 2 ) ∆ 3 + ( ∆ 3 ⊗ 1 + 1 ⊗ ∆ 3 ) ∆ 2 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 24 / 33

Operations on 2-cells of BR(3) Use Minnich’s formula to define ∆ k , k ≥ 3 , on each 2-cell, e.g., Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 25 / 33

Operations on 2-cells of BR(3) Use Minnich’s formula to define ∆ k , k ≥ 3 , on each 2-cell, e.g., ∆ 3 ( s 3 ) = c 15 ⊗ m 9 ⊗ c 5 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 25 / 33

Operations on 2-cells of BR(3) Use Minnich’s formula to define ∆ k , k ≥ 3 , on each 2-cell, e.g., ∆ 3 ( s 3 ) = c 15 ⊗ m 9 ⊗ c 5 ∆ 4 ( t 1 ) = m 11 ⊗ c 3 ⊗ c 4 ⊗ m 10 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 25 / 33

Operations on 2-cells of BR(3) Use Minnich’s formula to define ∆ k , k ≥ 3 , on each 2-cell, e.g., ∆ 3 ( s 3 ) = c 15 ⊗ m 9 ⊗ c 5 ∆ 4 ( t 1 ) = m 11 ⊗ c 3 ⊗ c 4 ⊗ m 10 ∆ 5 ( s 9 ) = m 11 ⊗ c 3 ⊗ m 13 ⊗ m 8 ⊗ c 13 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 25 / 33

Operations on 2-cells of BR(3) Use Minnich’s formula to define ∆ k , k ≥ 3 , on each 2-cell, e.g., ∆ 3 ( s 3 ) = c 15 ⊗ m 9 ⊗ c 5 ∆ 4 ( t 1 ) = m 11 ⊗ c 3 ⊗ c 4 ⊗ m 10 ∆ 5 ( s 9 ) = m 11 ⊗ c 3 ⊗ m 13 ⊗ m 8 ⊗ c 13 ∆ k = 0 on 2-cells for all k ≥ 6 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 25 / 33

Operations on 3-cells of BR(3) M. Fansler (2016) computed ∆ 3 on 3-cells, e.g., Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 26 / 33

Operations on 3-cells of BR(3) M. Fansler (2016) computed ∆ 3 on 3-cells, e.g., ∆ 3 ( q 1 ) = t 1 ⊗ m 12 ⊗ ( c 6 + m 8 + c 13 ) + t 1 ⊗ c 6 ⊗ ( m 8 + c 13 ) + t 1 ⊗ m 8 ⊗ c 13 + s 3 ⊗ c 6 ⊗ ( m 8 + c 13 ) + s 3 ⊗ m 8 ⊗ c 13 + s 7 ⊗ m 8 + c 13 + ( c 1 + c 15 ) ⊗ t 5 ⊗ c 13 + c 1 ⊗ s 3 ⊗ ( c 6 + m 8 + c 13 ) + c 3 ⊗ s 7 ⊗ ( m 8 + c 13 ) + m 11 ⊗ s 7 ⊗ ( m 8 + c 13 ) + c 1 ⊗ c 15 ⊗ t 5 + m 11 ⊗ c 3 ⊗ s 7 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 26 / 33

Operations on 3-cells of BR(3) M. Fansler (2016) computed ∆ 3 on 3-cells, e.g., ∆ 3 ( q 1 ) = t 1 ⊗ m 12 ⊗ ( c 6 + m 8 + c 13 ) + t 1 ⊗ c 6 ⊗ ( m 8 + c 13 ) + t 1 ⊗ m 8 ⊗ c 13 + s 3 ⊗ c 6 ⊗ ( m 8 + c 13 ) + s 3 ⊗ m 8 ⊗ c 13 + s 7 ⊗ m 8 + c 13 + ( c 1 + c 15 ) ⊗ t 5 ⊗ c 13 + c 1 ⊗ s 3 ⊗ ( c 6 + m 8 + c 13 ) + c 3 ⊗ s 7 ⊗ ( m 8 + c 13 ) + m 11 ⊗ s 7 ⊗ ( m 8 + c 13 ) + c 1 ⊗ c 15 ⊗ t 5 + m 11 ⊗ c 3 ⊗ s 7 ∆ 4 and ∆ 5 remains to be computed on 3-cells Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 26 / 33

Operations on 3-cells of BR(3) M. Fansler (2016) computed ∆ 3 on 3-cells, e.g., ∆ 3 ( q 1 ) = t 1 ⊗ m 12 ⊗ ( c 6 + m 8 + c 13 ) + t 1 ⊗ c 6 ⊗ ( m 8 + c 13 ) + t 1 ⊗ m 8 ⊗ c 13 + s 3 ⊗ c 6 ⊗ ( m 8 + c 13 ) + s 3 ⊗ m 8 ⊗ c 13 + s 7 ⊗ m 8 + c 13 + ( c 1 + c 15 ) ⊗ t 5 ⊗ c 13 + c 1 ⊗ s 3 ⊗ ( c 6 + m 8 + c 13 ) + c 3 ⊗ s 7 ⊗ ( m 8 + c 13 ) + m 11 ⊗ s 7 ⊗ ( m 8 + c 13 ) + c 1 ⊗ c 15 ⊗ t 5 + m 11 ⊗ c 3 ⊗ s 7 ∆ 4 and ∆ 5 remains to be computed on 3-cells ∆ k = 0 for all k ≥ 6 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 26 / 33

Introduction Transfer Algorithm Implementation Examples Conclusions Transferring Coproducts Goal: A ∞ -coalgebra on chains ( C , ∂, ∆ 2 , ∆ 3 , ... ) ↓ ( H , 0 , ∆ 2 , ∆ 3 , ... ) A ∞ -coalgebra in homology Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Transferring Coproducts Required input: Coalgebra on chains ( C , ∂, ∆ 2 , ∆ 3 , ... ) and a cycle-selecting map g : H → Z ( C ), where Z ( C ) denotes the subspace of cycles in C . Note: In practice we only required ∆ 2 at the outset and computed the rest as needed. Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions How Does It Work? Strategy: Construct a chain map from the top dimension and codim-1 cells of the ( n − 1)-dimensional multiplihedron, denoted J n , to maps between H and C ⊗ n . Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Beginning Steps J n is a polytope that captures the combinatiorial structure of mapping between two A ∞ -coalgebras. Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Beginning Steps J n is a polytope that captures the combinatiorial structure of mapping between two A ∞ -coalgebras. Consider J 1 and J 2 . Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Extending to J 3 ∆ 2 ⊗ 1 g ⊗ 3 � ∆ 2 g ⊗ 3 � 1 ⊗ ∆ 2 � ∆ 2 � g 2 ⊗ g ∆ 2 g ⊗ g 2 � ∆ 2 � � � (∆ 2 g ⊗ g ) ∆ 2 ( g ⊗ ∆ 2 g ) ∆ 2 �→ (∆ 2 ⊗ 1 ) g 2 ( 1 ⊗ ∆ 2 ) g 2 (∆ 2 ⊗ 1 ) ∆ 2 g ∆ 3 g ( 1 ⊗ ∆ 2 ) ∆ 2 g Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Table of Contents Introduction 1 Transfer Algorithm 2 Implementation 3 Examples 4 Conclusions 5 Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Linear Algebraic Methods Good News Linear algebra provides robust and theoretically correct methods for solving the various induction steps of the transfer algorithm. Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Linear Algebraic Methods Good News Bad News Linear algebra provides robust The matrices are too large to be and theoretically correct methods solved within a reasonable for solving the various induction amount of storage space and steps of the transfer algorithm. time. Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Two Problems Problem (Preboundary) Given a cycle x ∈ C ⊗ n of degree k, find a chain y ∈ C ⊗ n of degree k + 1 , such that ∂ ( y ) = x. Problem (Factorization) Given a cycle c ∈ Z ( C ⊗ n ) , find all subcycles of c of the form Z ( C ) ⊗ n . Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Preboundary Problem: ∆ 3 First problem arose in computing ∆ 3 Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Preboundary Problem: ∆ 3 First problem arose in computing ∆ 3 It is the preboundary of (∆ 2 ⊗ 1 + 1 ⊗ ∆ 2 )∆ 2 Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Preboundary Problem: ∆ 3 First problem arose in computing ∆ 3 It is the preboundary of (∆ 2 ⊗ 1 + 1 ⊗ ∆ 2 )∆ 2 Brute force linear algebra approach entails 1.8 mil row × 4 mil column matrix Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Preboundary Problem: ∆ 3 First problem arose in computing ∆ 3 It is the preboundary of (∆ 2 ⊗ 1 + 1 ⊗ ∆ 2 )∆ 2 Brute force linear algebra approach entails 1.8 mil row × 4 mil column matrix Instead, solved with a best-first search algorithm Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Factorization Problem Second problem comes from deriving ∆ n Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Factorization Problem Second problem comes from deriving ∆ n Transfer Algorithm specifies computing [ φ n ], i.e., H ∗ ( Hom ( H , Z ( C ⊗ ( n +2) ))) Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Factorization Problem Second problem comes from deriving ∆ n Transfer Algorithm specifies computing [ φ n ], i.e., H ∗ ( Hom ( H , Z ( C ⊗ ( n +2) ))) unneth Theorem tells us that H ∗ ( C ⊗ n ) ∼ However, K¨ = H ∗ ( C ) ⊗ n Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Factorization Problem Second problem comes from deriving ∆ n Transfer Algorithm specifies computing [ φ n ], i.e., H ∗ ( Hom ( H , Z ( C ⊗ ( n +2) ))) unneth Theorem tells us that H ∗ ( C ⊗ n ) ∼ However, K¨ = H ∗ ( C ) ⊗ n Hence, non-boundary cycles in φ n in should be of the form Z ( C ) ⊗ ( n +2) Merv Fansler Transfer Algorithm on BR3

Introduction Transfer Algorithm Implementation Examples Conclusions Factorization Problem Second problem comes from deriving ∆ n Transfer Algorithm specifies computing [ φ n ], i.e., H ∗ ( Hom ( H , Z ( C ⊗ ( n +2) ))) unneth Theorem tells us that H ∗ ( C ⊗ n ) ∼ However, K¨ = H ∗ ( C ) ⊗ n Hence, non-boundary cycles in φ n in should be of the form Z ( C ) ⊗ ( n +2) Again, an algorithmic approach appears to be a feasible alternative Merv Fansler Transfer Algorithm on BR3

Induced Operations Computed by M. Fansler H 0 = { 0 0 } , H 1 = { 1 0 , 1 1 , 1 2 } , H 2 = { 2 0 , 2 1 } Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 27 / 33

Induced Operations Computed by M. Fansler H 0 = { 0 0 } , H 1 = { 1 0 , 1 1 , 1 2 } , H 2 = { 2 0 , 2 1 } ∆ 2 ( 0 0 ) = 0 0 ⊗ 0 0 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 27 / 33

Induced Operations Computed by M. Fansler H 0 = { 0 0 } , H 1 = { 1 0 , 1 1 , 1 2 } , H 2 = { 2 0 , 2 1 } ∆ 2 ( 0 0 ) = 0 0 ⊗ 0 0 ∆ 2 ( 1 0 ) = 0 0 ⊗ 1 0 + 1 0 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 27 / 33

Induced Operations Computed by M. Fansler H 0 = { 0 0 } , H 1 = { 1 0 , 1 1 , 1 2 } , H 2 = { 2 0 , 2 1 } ∆ 2 ( 0 0 ) = 0 0 ⊗ 0 0 ∆ 2 ( 1 0 ) = 0 0 ⊗ 1 0 + 1 0 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ∆ 3 ( 2 0 ) = 1 0 ⊗ 1 1 ⊗ 1 2 + 1 0 ⊗ 1 2 ⊗ 1 1 + 1 1 ⊗ 1 2 ⊗ 1 0 + 1 2 ⊗ 1 1 ⊗ 1 0 ∆ 3 ( 2 1 ) = 1 0 ⊗ 1 1 ⊗ 1 2 + 1 1 ⊗ 1 0 ⊗ 1 2 + 1 2 ⊗ 1 0 ⊗ 1 1 + 1 2 ⊗ 1 1 ⊗ 1 0 Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 27 / 33

Induced Operations Computed by M. Fansler H 0 = { 0 0 } , H 1 = { 1 0 , 1 1 , 1 2 } , H 2 = { 2 0 , 2 1 } ∆ 2 ( 0 0 ) = 0 0 ⊗ 0 0 ∆ 2 ( 1 0 ) = 0 0 ⊗ 1 0 + 1 0 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ∆ 2 ( 1 1 ) = 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ∆ 3 ( 2 0 ) = 1 0 ⊗ 1 1 ⊗ 1 2 + 1 0 ⊗ 1 2 ⊗ 1 1 + 1 1 ⊗ 1 2 ⊗ 1 0 + 1 2 ⊗ 1 1 ⊗ 1 0 ∆ 3 ( 2 1 ) = 1 0 ⊗ 1 1 ⊗ 1 2 + 1 1 ⊗ 1 0 ⊗ 1 2 + 1 2 ⊗ 1 0 ⊗ 1 1 + 1 2 ⊗ 1 1 ⊗ 1 0 Linkage detected but ∆ 4 remains to be computed Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 27 / 33

The Case of BR(n) B. Nimershiem found an inductive way to construct a cellular decomposition of BR n Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 28 / 33

The Case of BR(n) B. Nimershiem found an inductive way to construct a cellular decomposition of BR n Her construction adjusts the decomposition of BR 3 so that all 2-cells have 5 edges Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 28 / 33

The Case of BR(n) B. Nimershiem found an inductive way to construct a cellular decomposition of BR n Her construction adjusts the decomposition of BR 3 so that all 2-cells have 5 edges Numbers of vertices, edges, faces, and solids in her decomposition are the same as in mine Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 28 / 33

Nimershiem’s Decomposition of BR(3) Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 29 / 33

Nimershiem’s Decomposition of BR(3) Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 30 / 33

Nimershiem’s Decomposition of BR(3) Dr. Ron Umble ( Millersville U and IMUS) Brunnian Links 2 May 2018 31 / 33