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Knot Invariant
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a ''knot invariant'' is a rule that assigns to any knot a quantity such that if and are equivalent then ."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification,Purcell, Jessica (2020). ''Hyperbolic Knot Theory'', p.7. American Mathematical Society. "A ''knot invariant'' is a function from the set of knots to some other set whose value depends only on the equiva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Table
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Khovanov Homology
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial. It was developed in the late 1990s by Mikhail Khovanov, then at the University of California, Davis, now at Columbia University. Overview To any link diagram ''D'' representing a link ''L'', we assign the Khovanov bracket ''D''.html" ;"title="/nowiki>''D''">/nowiki>''D''/nowiki>, a cochain complex of graded vector spaces. This is the analogue of the Kauffman bracket in the construction of the Jones polynomial. Next, we normalise ''D''.html" ;"title="/nowiki>''D''">/nowiki>''D''/nowiki> by a series of degree shifts (in the graded vector spaces) and height shifts (in the cochain complex) to obtain a new cochain complex C(''D''). The cohomology of this cochain complex turns out to be an invariant of ''L'', and its graded Euler characteristic is the Jones polynomial of ''L''. Definition This defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mostow Rigidity Theorem
Mostow may refer to: People * George Mostow (1923–2017), American mathematician ** Mostow rigidity theorem * Jonathan Mostow Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed films such as ''Breakdown (1997 film), Breakdown'', ''U-571 (film), U-571'', ''Terminator 3: Rise of the Machines'', and ''Surroga ... (born 1961), American movie and television director Places * Mostów, a village in Poland {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peripheral Subgroup
In algebraic topology, a peripheral subgroup for a topological pair, space-subspace pair ''X'' ⊃ ''Y'' is a certain subgroup of the fundamental group of the complementary space, π1(''X'' − ''Y''). Its conjugacy class is an invariant of the pair (''X'',''Y''). That is, any homeomorphism (''X'', ''Y'') → (''X''′, ''Y''′) induces an isomorphism π1(''X'' − ''Y'') → π1(''X''′ − ''Y''′) taking peripheral subgroups to peripheral subgroups. A peripheral subgroup consists of Loop (topology), loops in ''X'' − ''Y'' which are peripheral to ''Y'', that is, which stay "close to" ''Y'' (except when passing to and from the Pointed space, basepoint). When an ordered generating set of a group, set of generators for a peripheral subgroup is specified, the subgroup and generators are collectively called a peripheral system for the pair (''X'', ''Y''). Peripheral systems are used in knot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Quandle
In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams. While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group. History In 1943, Mituhisa Takasaki (高崎光久) introduced an algebraic structure which he called a ''Kei'' (圭), which would later come to be known as an involutive quandle. His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry. The idea was rediscovered and generalized in (unpublished) 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mirror Image (knot Theory)
In the mathematical field of knot theory, a chiral knot is a knot that is ''not'' equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible. There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.. Background The possible chirality of certain knots was suspected since 1847 when Johann Listing asserted that the trefoil was chiral, and this was proven by Max Dehn in 1914. P. G. Tait found all amphicheiral knots up to 10 crossings and conjectured that all amphicheiral knots had even crossing number. Mary Gertrude Haseman found all 12-crossing and many 14-crossing amphichei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gordon–Luecke Theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are '' isotopic''. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic. These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere. The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the proo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Complement
In mathematics, the knot complement of a tame knot ''K'' is the space where the knot is not. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that ''K'' is a knot in a three-manifold ''M'' (most often, ''M'' is the 3-sphere). Let ''N'' be a tubular neighborhood of ''K''; so ''N'' is a solid torus. The knot complement is then the complement of ''N'', :X_K = M - \mbox(N). The knot complement ''XK'' is a compact 3-manifold; the boundary of ''XK'' and the boundary of the neighborhood ''N'' are homeomorphic to a two-torus. Sometimes the ambient manifold ''M'' is understood to be 3-sphere. Context is needed to determine the usage. There are analogous definitions of link complement. Many knot invariants, such as the knot group, are really invariants of the complement of the knot. When the ambient space is the three-sphere no information is lost: the Gordon–Luecke theorem states that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutation (knot Theory)
In the mathematical field of knot theory, a mutation is an operation on a knot that can produce different knots. Suppose ''K'' is a knot given in the form of a knot diagram. Consider a disc ''D'' in the projection plane of the diagram whose boundary circle intersects ''K'' exactly four times. We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with ''K'' are equally spaced. The part of the knot inside the disc is a tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a mutant of ''K''. Mutants can be difficult to distinguish as they have a number of the same invariants. They have the same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomial In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Genus
In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research. Specifically, let ''L'' be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface ''S'' embedded in 3-space whose boundary is ''L'' such that the orientation on ''L'' is just the induced orientation from ''S''. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possible to ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
