Detecting Linkage in an n-Component Brunnian Link (work in progress) Ron Umble, Barbara Nimershiem, and Merv Fansler Millersville U and Franklin & Marshall College Tetrahedral Geometry/Topology Seminar December 4, 2015 Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 1/28 Ultimate Goal of the Project Computationally detect the linkage in an n-component Brunnian link Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 2/28 Review of Cellular Complexes Let X be a connected network, surface, or solid embedded in S3 Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 3/28 closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 non-empty intersection of cells is a cell union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 union of all cells is X Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 Cellular Decompositions A cellular decomposition of X is a finite collection of discrete points (vertices or 0-cells) closed intervals (edges or 1-cells) closed disks (faces or 2-cells) closed balls (solids or 3-cells) glued together so that the non-empty boundary of a k-cell is a union of (k 1)-cells non-empty intersection of cells is a cell union of all cells is X Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 4/28 Example: 2-dim’lSphere S2 = D2/¶D2 (Grandma’sdraw string bag) Vertex: v f g Edges: ? Face: S2 f g Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 5/28 Example: Torus T = S1 S1 Product cells: v,a v,b f g f g Vertex: v: = v v f g Edges: a := a v, b := v b f g Face: T := a b f g Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 6/28 Example: Pinched Sphere P = T /b Vertex: v f g Edge: a f g Face: S f g Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 7/28 Example: Link Complement of Two Unknots Let UN be the complement of disjoint tubular neighborhoods Ui of two unlinked unknots in S3 ¶ (U1 U2) is the wedge of two pinched spheres ti sharing a single vertex[v and two edges a and b ¶ (U1 U2) = ¶ (UN) [ Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 8/28 Cellular Structure of UN ¶ (UN) is wedged with the equatorial 2-sphere s S3 p = upper hemispherical 3-ball q = lower hemispherical 3-ball r (U1 U2) [ p and q are attached along s UN = p q [ Vertices: v f g Edges: a, b f g Faces: s, t1, t2 f g Solids: p, q f g Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 9/28 Example: Link Complement of the Hopf Link Let LN be the complement of disjoint tubular neighborhoods Ui of the Hopf Link in S3 ¶ (U1 U2) is the union of two linked tori ti0 sharing a single vertex v and two[ edges a and b ¶ (U1 U2) = ¶ (LN) [ Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 10/28 Cellular Structure of LN ¶ (LN) is wedged with the equatorial 2-sphere s S3 p = upper hemispherical 3-ball q0 = lower hemispherical 3-ball r (U1 U2) [ p and q0 are attached along s LN = p q [ 0 Vertex: v Edges: fa, bg f g Faces: s, t10 , t20 Solids: f p, q g f 0g Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 11/28 a bijective bicontinuous h : X Y , called a homeomorphism 9 ! A square and a circle are homeomorphic h ! The boundaries of a doughnut and coffee cup are homeomorphic Homeomorphisms X and Y are homeomorphic if X can be continuously deformed into Y Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 12/28 A square and a circle are homeomorphic h ! The boundaries of a doughnut and coffee cup are homeomorphic Homeomorphisms X and Y are homeomorphic if X can be continuously deformed into Y a bijective bicontinuous h : X Y , called a homeomorphism 9 ! Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 12/28 The boundaries of a doughnut and coffee cup are homeomorphic Homeomorphisms X and Y are homeomorphic if X can be continuously deformed into Y a bijective bicontinuous h : X Y , called a homeomorphism 9 ! A square and a circle are homeomorphic h ! Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 12/28 Homeomorphisms X and Y are homeomorphic if X can be continuously deformed into Y a bijective bicontinuous h : X Y , called a homeomorphism 9 ! A square and a circle are homeomorphic h ! The boundaries of a doughnut and coffee cup are homeomorphic Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 12/28 shrinks ¶ (UN) to a pinched sphere shrinks ¶ (LN) to a 2-sphere Shrinking the tubular neighborhood of one component to point How do we can detect this computationally? Homeomorphisms UN and LN are not homeomorphic because... Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 13/28 shrinks ¶ (UN) to a pinched sphere shrinks ¶ (LN) to a 2-sphere How do we can detect this computationally? Homeomorphisms UN and LN are not homeomorphic because... Shrinking the tubular neighborhood of one component to point Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 13/28 shrinks ¶ (LN) to a 2-sphere How do we can detect this computationally? Homeomorphisms UN and LN are not homeomorphic because... Shrinking the tubular neighborhood of one component to point shrinks ¶ (UN) to a pinched sphere Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 13/28 How do we can detect this computationally? Homeomorphisms UN and LN are not homeomorphic because... Shrinking the tubular neighborhood of one component to point shrinks ¶ (UN) to a pinched sphere shrinks ¶ (LN) to a 2-sphere Tetrahedral Geometry/Topology Seminar ( DetectingTetrahedral Brunnian Geometry/Topology Linkage Seminar ) December4,2015 13/28 Homeomorphisms UN and LN are not homeomorphic because... Shrinking the tubular neighborhood of one component to point shrinks ¶ (UN) to a pinched sphere shrinks ¶ (LN) to a 2-sphere How do we can detect this computationally? Tetrahedral Geometry/Topology