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Way to join two given mathematical manifolds together
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds
Connected_sum
Gluing graphs at complete subgraphs
mathematics, a clique sum (or clique-sum) is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology
Clique-sum
Topological invariant in mathematics
alternating sum of ranks of reduced homology groups. For two connected closed n-manifolds M , N {\displaystyle M,N} one can obtain a new connected manifold
Euler_characteristic
Two-dimensional manifold
is T # M. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined. The connected sum of two real projective
Surface_(topology)
Type of topological space
2-manifold (or surface) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes. A classification of 3-manifolds
Topological_manifold
Connected sum of two trefoil knots with opposite chirality
by taking the connected sum of a trefoil knot with its reflection. It is closely related to the granny knot, which is also a connected sum of two trefoils
Square_knot_(mathematics)
It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of the symplectic
Symplectic_sum
Topics referred to by the same term
theory Band sum, a way of connecting mathematical knots Connected sum, a way of gluing manifolds Digit sum, in number theory Direct sum, a combination
Sum
Three dimensional analogue of uniformization conjecture
Every closed 3-manifold has a prime decomposition: this means it is the connected sum ("a gluing together") of prime 3-manifolds. This reduces much of the
Geometrization_conjecture
Smooth closed surface with g holes
(also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g
Genus_g_surface
Smooth manifold that is homeomorphic but not diffeomorphic to a sphere
exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification
Exotic_sphere
Space in topology mathematics
{\displaystyle Y.} Connected sum Smash product Hawaiian earring, a topological space resembling, but not the same as, a wedge sum of countably many circles
Wedge_sum
Connected sum of two trefoil knots with same chirality
taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils
Granny_knot_(mathematics)
On structure of ω-bounded connected surfaces
structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag"
Bagpipe_theorem
Mathematical space
cannot be described as a connected sum of two 3-manifolds is called prime. For the case of a 3-manifold given by a connected sum of prime 3-manifolds, it
3-manifold
Method of connecting knots
n-dimensional knot obtained by this surgery. A band sum is thus a generalization of the usual connected sum of knots. Manifold decomposition Cromwell, Peter
Band_sum
3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds
Prime decomposition of 3-manifolds
Prime_decomposition_of_3-manifolds
Mathematical concept
kept) Connected space Connected category Connected component (graph theory) Connected sum Cross-link Network Scale-free network Simply connected Small-world
Connectedness
Doughnut-shaped surface of revolution
form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense
Torus
Abelian group, in mathematics
group defined as the h-cobordism classes of homotopy spheres with the connected sum as composition and the reverse orientation as inversion. It controls
Kervaire–Milnor_group
N_{1}\#N_{2}} with their connected sum. (Alternatively M ≅ ∗ N 1 # ∗ N 2 {\displaystyle M\cong *N_{1}\#*N_{2}} .) For every simply connected oriented closed topological
Freedman_classification
Mathematical knot with crossing number 7
counterexample to the conjecture that the unknotting number is additive under connected sum. The 71 knot is invertible but not amphichiral. Its Alexander polynomial
71_knot
Number divisible only by 1 and itself
it cannot be written as the connected sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots. The prime decomposition
Prime_number
Topological manifold whose homology coincides with that of a sphere
sphere. The connected sum of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology
Homology_sphere
Branch of topology
states that any connected closed surface is homeomorphic to some member of one of these three families: the sphere; the connected sum of g tori, for g
Low-dimensional_topology
Topics referred to by the same term
connection, a transfer from one means of transport to another Connected sum Connectedness Connecting (TV series) Connections (disambiguation) Connexion
Connection
Types of electrical circuits
across the network is equal to the sum of the voltages across each component. Components connected in parallel are connected along multiple paths, and each
Series_and_parallel_circuits
Topological manifold in mathematics
yields the following three orientation-preserving homeomorphisms between connected sums of the E 8 {\displaystyle E_{8}} -manifold M E 8 {\displaystyle M_{E_{8}}}
E8_manifold
Immersion of the Klein bottle
Topologically, the Ida surface is the connected sum of two copies of Boy's surface, just as the Klein bottle is the connected sum of two copies of the real projective
Ida_surface
Type of n-manifold in topology
manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an
Prime_manifold
Minimum number of times a specific knot must be passed through itself to become untied
showed that the unknotting number of the connected sum of 71 and its mirror image was at most 5, one less than the sum of the numbers from its components.
Unknotting_number
Concept in algebraic topology
abelian group known as Kervaire–Milnor group. Its composition is the connected sum and its neutral element is the sphere, while inversion is given by opposite
Homotopy_sphere
Smooth 4-manifold homeomorphic yet not diffeomorphic to Euclidean space
According to Freedman's classification, it is homeomorphic to the connected sum 2 M E 8 ¯ # 3 ( S 2 × S 2 ) {\displaystyle 2{\overline {M_{E_{8}}}}\#3(S^{2}\times
Exotic_R4
Method of proof involving paradoxical properties of infinite sums
swindle is usually the connected sum of knots or manifolds. A typical application of the Mazur swindle is the proof that the sum of two non-trivial knots
Eilenberg–Mazur_swindle
Self-intersecting compact surface, an immersion of the real projective plane
points, is topologically the connected sum of two copies of Boy's surface, just as the Klein bottle is the connected sum of two copies of the real projective
Boy's_surface
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
Schubert (1919–2001) states that every knot can be uniquely expressed as a connected sum of prime knots. List of prime knots Thistlethwaite, M. "The enumeration
Prime_knot
Knot that bounds an embedded disk in 4-space
K 2 {\displaystyle K_{1},K_{2}} are said to be concordant, if the connected sum K 1 ♯ − K 2 {\displaystyle K_{1}\sharp -K_{2}} is slice. In the same
Slice_knot
Study of mathematical knots
joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined
Knot_theory
Topics referred to by the same term
Square knot (mathematics), a composite knot obtained by taking the connected sum of a trefoil knot with its reflection Square knot insignia, embroidered
Square_knot
Russian mathematician (born 1966)
manifolds with positive Ricci curvature. He found Riemannian metrics on the connected sum of arbitrarily many complex projective planes with positive Ricci curvature
Grigori_Perelman
Measure of curvature in differential geometry
the connected sum. This establishes the existence of such metrics on a wide variety of manifolds. For example, it immediately shows that the connected sum
Scalar_curvature
Specific element of an algebraic structure
(nonequivalence) ⊥ {\textstyle \bot } (falsity) Knots Knot sum Unknot Compact surfaces # (connected sum) S2 Abstract groups Direct product Trivial group Two
Identity_element
Algebraic structure associated with a topological space
realised as the connected sum of g tori and c projective planes, where the 2-sphere S 2 {\displaystyle S^{2}} is regarded as the empty connected sum. Homology
Homology_(mathematics)
Non-orientable mathematical surface
theorem, which would require seven. A Klein bottle is homeomorphic to the connected sum of two projective planes. It is also homeomorphic to a sphere plus two
Klein_bottle
Ancient Mesopotamian civilization from 3300 to 1900 BC
Sumer (/ˈsuːmər/ SOO-mər) is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging
Sumer
Undirected, connected, and acyclic graph
in which every pair of distinct vertices is connected by exactly one path, or equivalently, a connected acyclic undirected graph. A forest is an undirected
Tree_(graph_theory)
Index of articles associated with the same name
related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible
Irreducibility_(mathematics)
Ways of coupling springs in mechanics
in series when they are connected end-to-end or point to point, and they are said to be in parallel when they are connected side-by-side; in both cases
Series_and_parallel_springs
Typographic symbol (#)
symbol, e.g. a ∣ b {\displaystyle a\mid b} . In topology, A#B is the connected sum of manifolds A and B, or of knots A and B in knot theory. In number
Number_sign
Invariant of a quadratic form over a field of characteristic 2
0}} direct summand), and so is a knot invariant. It is additive under connected sum, and vanishes on slice knots, so is a knot concordance invariant. The
Arf_invariant
Sum in algebraic number theory
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G ( χ ) := G ( χ , ψ ) = ∑ χ (
Gauss_sum
Algebraic surface defined by a cubic polynomial
smooth cubic surface over the complex numbers is diffeomorphic to the connected sum C P 2 # 6 ( − C P 2 ) {\displaystyle \mathbf {CP} ^{2}\#6(-\mathbf {CP}
Cubic_surface
and diffeomorphisms; counterexamples can be constructed by taking a connected sum with an exotic sphere. In a May 1953 letter to Jean-Pierre Serre, Armand
Borel_conjecture
Type of planar curve with tree-like structure
knot diagrams, these represent connected sums of figure-eight curves. Each figure-eight is unknotted and their connected sum remains unknotted. Random curves
Tree-like_curve
American mathematician
that the unknotting number is not additive under the connected sum. They proved that the connected sum of the Septoil knot with its mirror image, 7 1 # 7
Susan_Hermiller
Topological spaces whose union is a boundary
cobordant to the connected sum M # M ′ . {\displaystyle M\mathbin {\#} M'.} The previous example is a particular case, since the connected sum S 1 # S 1 {\displaystyle
Cobordism
Connected sum of copies of the complex projective plane
In mathematics, a LeBrun manifold is a connected sum of copies of the complex projective plane, equipped with an explicit self-dual metric. Here, self-dual
Lebrun_manifold
4-manifold invariants
the connected sum of two manifolds both of which have b2+ ≥ 1 then all Seiberg–Witten invariants of M vanish. If the manifold M is simply connected and
Seiberg–Witten_invariants
greater than n. 3. In topology, M # N {\displaystyle M\#N} denotes the connected sum of two manifolds or two knots. ∈ Denotes set membership, and is read
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Graph topology applied to electrical and communications circuits, or biomolecules
their relationship. Knot theory considers any entangled chain as a connected sum of prime knots, which are themselves undecomposable. Circuit topology
Circuit_topology
Theorem in vector calculus
{\displaystyle \Gamma } is always a loop or loops, and topologically a connected sum of countably many Jordan curves, so that the integrals are well-defined
Stokes'_theorem
Analyzes the topology of a manifold by studying differentiable functions on that manifold
, {\displaystyle g>0,} M {\displaystyle M} is diffeomorphic to the connected sum of g {\displaystyle g} 2-tori. If N {\displaystyle N} is unorientable
Morse_theory
Mathematical formula of two surfaces
from a surface lowers its Euler characteristic by 1 by the formula for connected sum, so we finish by the formula for a non-ramified covering. We can also
Riemann–Hurwitz_formula
taking the connected sum of a trefoil knot with its reflection Granny knot (mathematics) - a composite knot obtained by taking the connected sum of two identical
List of mathematical knots and links
List_of_mathematical_knots_and_links
Algebraic structure with an associative operation and an identity element
below. The set of homeomorphism classes of compact surfaces with the connected sum. Its unit element is the class of the ordinary 2-sphere. Furthermore
Monoid
Theorem in group theory
which says that a closed 3-manifold can be uniquely decomposed as a connected sum of irreducible 3-manifolds. The following is a sketch of the proof of
Grushko_theorem
Two embeddings of a closed k-disc into a connected n-manifold are ambient isotopic
connected n-manifold are ambient isotopic provided that if k = n the two embeddings are equioriented. The disc theorem implies that the connected sum
Disc_theorem
Natural number
sum of four consecutive primes (167 + 173 + 179 + 181). 701 is a prime number, a Chen prime, an Eisenstein prime with no imaginary part, and the sum of
700_(number)
Mathematical notation for describing the structure of knots
differ from the original by either being a reflection or by having any connected sum component reflected in the line between its entry/exit points – the
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite_notation
Study of the topology of a complex manifold
degenerations whenever t = a i {\displaystyle t=a_{i}} . Since the curve is a connected sum of g {\displaystyle g} tori, the intersection form on H 1 {\displaystyle
Picard–Lefschetz_theory
Family of quantum invariants
M){\text{RT}}_{r}(N),} where M # N {\displaystyle M\#N} denotes the connected sum of M {\displaystyle M} and N {\displaystyle N} RT r ( − M ) = RT r
Reshetikhin–Turaev_invariant
knots are rational knots. If K is the connected sum of K1 and K2, then the bridge number of K is one less than the sum of the bridge numbers of K1 and K2
Bridge_number
Path in a graph that visits each vertex exactly once
the sum of their degrees is n or greater. The following theorems can be regarded as directed versions: Ghouila–Houiri (1960)—A strongly connected simple
Hamiltonian_path
or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams (the sum over
Ursell_function
Mathematical space
|signature|), there is a smooth structure: the manifold is homeomorphic to a connected sum of n K3 surfaces and m − 3n copies of S2×S2. For m ≤ 2n (so the dimension
4-manifold
Figurate number
2007. Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function
Triangular_number
Group of isotopy classes of a topological automorphism group
also remark that the closed genus three non-orientable surface N3 (the connected sum of three projective planes) has: MCG ( N 3 ) = GL ( 2 , Z ) . {\displaystyle
Mapping_class_group
by adding a constant to each strand). If # {\displaystyle \#} is the connected sum operator and L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}}
Fox_n-coloring
Regularization method for artificial neural networks
fixed fraction of the weights are diluted. When the number of terms in the sum goes to infinite (the weights for each node) it is still infinite (the fraction
Dropout_(neural_networks)
Results in differential topology
S^{2})\cong N\#n(S^{2}\times S^{2})} . Becoming diffeomorphic after connected sums with S 2 × S 2 {\displaystyle S^{2}\times S^{2}} is also called stably
Wall_theorems
period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with
Gaussian_period
Decomposition of a manifold into standard pieces
torus. All other handlebodies may be obtained by taking the boundary-connected sum of a collection of solid tori. Handle decomposition Matsumoto, Yukio
Handlebody
Binary operation combining the vertex and edge sets of two graphs
is necessarily disconnected. The disjoint union is also called the graph sum and may be represented either by a plus sign or a circled plus sign: If G
Disjoint_union_of_graphs
Supplementary pair of angles at each vertex of a polygon
triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on. The sum of the external angles of
Internal_and_external_angles
American mathematician
of knot invariants like bridge number when knots are combined by the connected sum operation, and the Kakimizu complexes of knot complements and other
Jennifer_Schultens
1991 thriller novel by Tom Clancy
The Sum of All Fears is a political thriller novel, written by Tom Clancy and released on August 14, 1991, as the sequel to Clear and Present Danger (1989)
The_Sum_of_All_Fears
which take the value j. n j = ∑ i = 1 N ( q i = j ) {\displaystyle n_{j}=\sum _{i=1}^{N}(q_{i}=j)} The total number of permutation of q → {\displaystyle
Pixel_connectivity
Maths
sphere on which a connected sum has been done – but need not, such as cutting along a curve on the torus. Cutting along a (connected) 1-sided manifold
2-sided
Conjecture in symplectic geometry
called stable if whenever it may be written as a graded Lagrangian connected sum ( L , ϑ ) = ( L 1 , ϑ 1 ) # ( L 2 , ϑ 2 ) {\displaystyle (L,\vartheta
Thomas–Yau_conjecture
dimension of the ambient space minus the dimension of the submanifold. Connected sum Connection Cotangent bundle – the vector bundle of cotangent spaces
Glossary of differential geometry and topology
Glossary_of_differential_geometry_and_topology
Mathematical construct
sum is a sum ∑ χ ( n ) {\textstyle \sum \chi (n)} of values of a Dirichlet character χ modulo N, taken over a given range of values of n. Such sums are
Character_sum
Type of geometry in mathematics
closed under homotopy equivalence, the taking of products, and under the connected sum with an arbitrary closed manifold. Every Ricci-flat Riemannian manifold
Ricci-flat_manifold
Type of smooth complex surface of kodaira dimension 0
would imply that every simply connected smooth 4-manifold with even intersection form is homeomorphic to a connected sum of copies of the K3 surface and
K3_surface
Special tangential structure
{\displaystyle N} are spinh manifolds of same dimension, then their connected sum M # N {\displaystyle M\#N} is a spinh manifold. The following conditions
Spinh_structure
Every graph has evenly many odd vertices
equals the sum of the cardinalities of the sets. Both results also apply to any subgraph of the given graph and in particular to its connected components
Handshaking_lemma
Decomposition of a compact oriented 3-manifold by dividing it into two handlebodies
Heegaard splitting H in M the stabilization of H is formed by taking the connected sum of the pair ( M , H ) {\displaystyle (M,H)} with the pair ( S 3 , T
Heegaard_splitting
Link formed from a finite number of twisted sections
unknots. The (−3, 0, −3) pretzel knot (square knot (mathematics)) is the connected sum of two trefoil knots. The (0, q, 0) pretzel link is the split union
Pretzel_link
Mathematical concept
} In dimension two, this yields a torus, and taking the connected sum of g tori yields the surface of genus g, whose middle homology has the
Ε-quadratic_form
Special tangential structure
{\displaystyle N} are spinc manifolds of same dimension, then their connected sum M # N {\displaystyle M\#N} is a spinc manifold. The following conditions
Spinc_structure
Mathematical property
Sol and Nil geometries, as well as 3-manifold groups obtained by some connected sum and JSJ decomposition constructions. For every n ≥ 1 {\displaystyle
Howson_property
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