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Slice Knot
A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the boundary of an Embedding, embedded disk in the 4-ball B^4, which is Local flatness, locally flat or Smoothness, smooth, respectively. Here we use S^3 = \partial B^4: the 3-sphere S^3 = \ is the boundary (topology), boundary of the four-dimensional ball (mathematics), ball B^4 = \. Every smoothly slice knot is topologically slice because a smoothly embedded disk is locally flat. Usually, smoothly slice knots are also just called slice. Both types of slice knots are important in 3- and 4-dimensional topology. Smoothly slice knots are often illustrated using knots diagrams of ribbon knots and it is an open question whether there are any smoothly slice knots which are not ribbon knots (′Slice-ribbon conjecture′). Cone construction The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Flatness
In topology, a branch of mathematics, local flatness is a smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Violations of local flatness describe ridge networks and crumpled structures, with applications to materials processing and mechanical engineering. Definition Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If x \in N, we say ''N'' is locally flat at ''x'' if there is a neighborhood U \subset M of ''x'' such that the topological pair (U, U\cap N) is homeomorphic to the pair (\mathbb^n,\mathbb^d), with the standard inclusion of \mathbb^d\to\mathbb^n. That is, there exists a homeomorphism U\to \mathbb^n such that the image of U\cap N coincides with \mathbb^d. In diagrammatic terms, the following square must commute: We c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slice Disk Annotated In English
Slice may refer to: *Cutting Food and beverage *A portion of bread, pizza, cake, meat, or other food item that is cut flat and thin: :*Sliced bread :*Pizza by the slice, a fast food dish :*Vanilla slice :*Fudge :*Toffee :*Barfi *Slice (drink), a line of fruit-flavored soft drinks *A serving or cooking utensil, such as a Fish slice In Australia and New Zealand *A category of sweet or savory dishes: :*Vanilla slice, a dessert cake similar to a brownie :*Zucchini slice, a savory dish similar to a quiche In arts and entertainment Music * ''Slice'' (album), a Five for Fighting album, 2009 ** "Slice" (song), a 2009 song by Five for Fighting *''Slice'', a 1998 album by Arthur Loves Plastic * Slices (band) * ''Slice'' (EP), 2023 by O. Other uses in arts and entertainment *Slice (TV channel), a Canadian TV channel formerly known as Life Network *Slice (2009 film), a Thai crime film * ''Slice'' (2018 film), an American horror comedy film *Slice (G.I. Joe), a fictional character in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Knot
In mathematics, specifically in knot theory, the Conway knot (or Conway's knot) is a particular knot (mathematics), knot with 11 crossings, named after John Horton Conway. It is related by mutation (knot theory), mutation to the Kinoshita–Terasaka knot, with which it shares the same Jones polynomial. Both knots also have the curious property of having the same Alexander polynomial and Alexander polynomial#Alexander–Conway polynomial, Conway polynomial as the unknot. The issue of the slice knot, sliceness of the Conway knot was resolved in 2020 by Lisa Piccirillo, 50 years after John Horton Conway first proposed the knot. Her proof made use of Khovanov homology#Applications, Rasmussen's s-invariant, and showed that the knot is not a smoothly slice knot, though it is topologically slice (the Kinoshita–Terasaka knot is both). References External links Conway knoton The Knot Atlas. Conway knot illustrated bknotilus . {{Knot theory, state=collapsed John Horton Conw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" and () meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German meaning "similar" to meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature Of A Knot
The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface. Given a knot ''K'' in the 3-sphere, it has a Seifert surface ''S'' whose boundary is ''K''. The Seifert form of ''S'' is the pairing \phi : H_1(S) \times H_1(S) \to \mathbb Z given by taking the linking number \operatorname(a^+,b^-) where a, b \in H_1(S) and a^+, b^- indicate the translates of ''a'' and ''b'' respectively in the positive and negative directions of the normal bundle to ''S''. Given a basis b_1,...,b_ for H_1(S) (where ''g'' is the genus of the surface) the Seifert form can be represented as a ''2g''-by-''2g'' Seifert matrix ''V'', V_=\phi(b_i,b_j). The signature of the matrix V+V^t, thought of as a symmetric bilinear form, is the signature of the knot ''K''. Slice knots are known to have zero signature. The Alexander module formulation Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let X be the universa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes. Early life and career Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor, an engineer, and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Ralph Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ralph Fox
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the ''Golden Age of differential topology'', and he played an important role in the modernization of knot theory and of bringing it into the mainstream. Biography Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a PhD degree from Princeton University in 1939. His doctoral dissertation, ''On the Lusternick–Schnirelmann Category'', was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Polynomial
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in X form a ring denoted \mathbb , X^/math>. They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables. Definition A Laurent polynomial with coefficients in a field \mathbb is an expression of the form : p = \sum_k p_k X^k, \quad p_k \in \mathbb where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients p_ are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial. Definition Let ''K'' be a knot in the 3-sphere. Let ''X'' be the infinite cyclic cover of the knot complement of ''K''. This covering can be obtained by cutting the knot complement along a Seifert surface of ''K'' and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation ''t'' acting on ''X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cameron Gordon (mathematician)
Cameron Gordon (born 1945) is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of Mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture. Andrew Casson and Gordon defined and proved basic theorems regarding strongly irreducible Heegaard splittings, an important concept in the modernization of Heegaard splitting theory. They also worked on the slice-ribbon conjecture, inventing the Casson-Gordon invariants in the process. Gordon was a 1999 Guggenheim Fellow. In 2005 Gordon was elected a Corresponding Fellow of the Royal Society of Edinburgh. In 2023 Gordon was elected to the National Academy of Sciences. References External li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrew Casson
Andrew John Casson FRS (born 1943) is a mathematician, studying geometric topology. Casson is the Philip Schuyler Beebe Professor of Mathematics at Yale University. Education and career Casson was educated at Latymer Upper School and Trinity College, Cambridge, where he graduated with a BA in the Mathematical Tripos in 1965.'University News: Cambridge Tripos Results', ''Times'', 21 June 1965. His doctoral advisor at the University of Liverpool was C. T. C. Wall, but he never completed his doctorate; instead what would have been his Ph.D. thesis became his fellowship dissertation as a research fellow at Trinity College. Casson was Professor of Mathematics at the University of Texas at Austin between 1981 and 1986, at the University of California, Berkeley, from 1986 to 2000, and has been at Yale since 2000. Work Casson has worked in both high-dimensional manifold topology and 3- and 4-dimensional topology, using both geometric and algebraic techniques. Among other discoveries, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stevedore Knot (mathematics)
A dockworker (also called a longshoreman, stevedore, docker, wharfman, lumper or wharfie) is a waterfront manual laborer who loads and unloads ships. As a result of the intermodal shipping container revolution, the required number of dockworkers has declined by over 90% since the 1960s. Etymology The word ''stevedore'' () originated in Portugal or Spain, and entered the English language through its use by sailors. It started as a phonetic spelling of ''estivador'' ( Portuguese) or ''estibador'' (Spanish), meaning ''a man who loads ships and stows cargo'', which was the original meaning of ''stevedore'' (though there is a secondary meaning of "a man who stuffs" in Spanish); compare Latin ''stīpāre'' meaning ''to stuff'', as in ''to fill with stuffing''. In Ancient and Modern Greek, the verb στοιβάζω (stivazo) means pile up. In Great Britain and Ireland, people who load and unload ships are usually called ''dockers''; in Australia, they are called ''stevedores'', ''dock ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
