|
|
| Line 1: |
Line 1: |
| The '''signature of a knot''' is a [[topological invariant]] in [[knot theory]]. It may be computed from the [[Seifert surface]].
| | Another day I woke up and realized - I have also been solitary for a little while at the moment and following much intimidation from friends I today find myself [http://www.ffpjp24.org luke bryan vip tickets for sale] signed-up for web dating. They guaranteed me that there are lots of entertaining, sweet and regular individuals to fulfill, so here goes the message!<br>My buddies and family are amazing and hanging out together at tavern gigs or meals is always vital. As I see that you can not own a good conversation against the noise I haven't ever been into clubs. I also have two quite adorable and definitely cheeky puppies that are constantly keen to meet fresh individuals.<br>I try and maintain as physically healthy as possible coming to [http://lukebryantickets.sgs-suparco.org luke bryan tickets cheap] the fitness center many times weekly. I love my [http://www.Google.de/search?q=athletics athletics] where can i buy luke bryan tickets, [http://lukebryantickets.asiapak.net http://lukebryantickets.asiapak.net], and attempt to perform or watch as numerous a potential. I will often at [http://www.wonderhowto.com/search/Hawthorn+matches/ Hawthorn matches] being wintertime. Note: I have noticed the carnage of fumbling matches at stocktake sales, In case that you really considered buying a hobby I really do not brain.<br><br> |
|
| |
|
| Given a [[knot (mathematics)|knot]] ''K'' in the [[3-sphere]], it has a [[Seifert surface]] ''S'' whose boundary is ''K''. The '''[[Seifert form]]''' of ''S'' is the pairing <math>\phi : H_1(S) \times H_1(S) \to \mathbb Z</math> given by taking the [[linking number]] <math>lk(a^+,b^-)</math> where <math>a, b \in H_1(S)</math> and <math>a^+, b^-</math> indicate the translates of ''a'' and ''b'' respectively in the positive and negative directions of the [[normal bundle]] to ''S''.
| | My web-site :: luke bryan tickets [http://lukebryantickets.hamedanshahr.com luke bryan concert dates 2014] cheap; [http://www.cinemaudiosociety.org mouse click the next internet page], |
| | |
| Given a basis <math>b_1,...,b_{2g}</math> for <math>H_1(S)</math> (where ''g'' is the genus of the surface) the Seifert form can be represented as a ''2g''-by-''2g'' [[Seifert matrix]] ''V'', <math>V_{ij}=\phi(b_i,b_j)</math>. The [[Symmetric bilinear form|signature]] of the matrix <math>V+V^\perp</math>, thought of as a symmetric bilinear form, is the signature of the knot ''K''.
| |
| | |
| [[Slice knot]]s are known to have zero signature.
| |
| | |
| ==The Alexander module formulation==
| |
| Knot signatures can also be defined in terms of the [[Alexander polynomial|Alexander module]] of the knot complement. Let <math>X</math> be the universal abelian cover of the knot complement. Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: <math>H_1(X;\mathbb Q)</math>. Given a <math>\mathbb Q[\mathbb Z]</math>-module <math>V</math>, let <math>\overline{V}</math> denote the <math>\mathbb Q[\mathbb Z]</math>-module whose underlying <math>\mathbb Q</math>-module is <math>V</math> but where <math>\mathbb Z</math> acts by the inverse covering transformation. Blanchfield's formulation of [[Poincaré duality]] for <math>X</math> gives a canonical isomorphism <math>H_1(X;\mathbb Q) \simeq \overline{H^2(X;\mathbb Q)}</math> where <math>H^2(X;\mathbb Q)</math> denotes the 2nd cohomology group of <math>X</math> with compact supports and coefficients in <math>\mathbb Q</math>. The universal coefficient theorem for <math>H^2(X;\mathbb Q)</math> gives a canonical isomorphism with <math>Ext_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),\mathbb Q[\mathbb Z])</math> (because the Alexander module is <math>\mathbb Q[\mathbb Z]</math>-torsion). Moreover, just like in the [[Poincaré duality|quadratic form formulation of Poincaré duality]], there is a canonical isomorphism of <math>\mathbb Q[\mathbb Z]</math>-modules <math> Ext_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),\mathbb Q[\mathbb Z]) \simeq Hom_{\mathbb Q[\mathbb Z]}(H_1(X;\mathbb Q),[\mathbb Q[\mathbb Z]]/\mathbb Q[\mathbb Z] )</math>, where <math>[\mathbb Q[\mathbb Z]]</math> denotes the field of fractions of <math>\mathbb Q[\mathbb Z]</math>. This isomorphism can be thought of as a sesquilinear duality pairing <math>H_1(X;\mathbb Q) \times H_1(X;\mathbb Q) \to [\mathbb Q[\mathbb Z]]/\mathbb Q[\mathbb Z]</math> where <math>[\mathbb Q[\mathbb Z]]</math> denotes the field of fractions of <math>\mathbb Q[\mathbb Z]</math>. This form takes value in the rational polynomials whose denominators are the [[Alexander polynomial]] of the knot, which as a <math>\mathbb Q[\mathbb Z]</math>-module is isomorphic to <math>\mathbb Q[\mathbb Z]/\Delta K</math>. Let <math>tr : \mathbb Q[\mathbb Z]/\Delta K \to \mathbb Q</math> be any linear function which is invariant under the involution <math>t \longmapsto t^{-1}</math>, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on <math>H_1 (X;\mathbb Q)</math> whose signature is an invariant of the knot.
| |
| | |
| All such signatures are concordance invariants, so all signatures of [[slice knot]]s are zero. The sesquilinear duality pairing respects the prime-power decomposition of <math>H_1 (X;\mathbb Q)</math> -- ie: the prime power decomposition gives an orthogonal decomposition of <math>H_1 (X;\mathbb R)</math>. Cherry Kearton has shown how to compute the ''Milnor signature invariants'' from this pairing, which are equivalent to the ''Tristram-Levine invariant''.
| |
| | |
| ==References==
| |
| * C.Gordon, Some aspects of classical knot theory. Springer Lecture Notes in Mathematics 685. Proceedings Plans-sur-Bex Switzerland 1977.
| |
| | |
| * J.Hillman, Algebraic invariants of links. Series on Knots and everything. Vol 32. World Scientific.
| |
| | |
| * C.Kearton, Signatures of knots and the free differential calculus, Quart. J. Math. Oxford (2), 30 (1979).
| |
| | |
| * J.Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44, 229-244 (1969)
| |
| | |
| * J.Milnor, Infinite cyclic coverings, J.G. Hocking, ed. Conf. on the Topology of Manifolds, Prindle, Weber and Schmidt, Boston, Mass, 1968 pp. 115–133.
| |
| | |
| * K.Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117, 387-482 (1965)
| |
| | |
| * A.Ranicki [http://www.maths.ed.ac.uk/~aar/slides/durham.pdf On signatures of knots] Slides of lecture given in Durham on 20 June 2010.
| |
| | |
| * H.Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76, 464-498 (1962)
| |
| | |
| ==See also==
| |
| *[[Link concordance]]
| |
| | |
| {{Use dmy dates|date=September 2010}}
| |
| | |
| {{DEFAULTSORT:Signature Of A Knot}}
| |
| [[Category:Knot theory]]
| |
Another day I woke up and realized - I have also been solitary for a little while at the moment and following much intimidation from friends I today find myself luke bryan vip tickets for sale signed-up for web dating. They guaranteed me that there are lots of entertaining, sweet and regular individuals to fulfill, so here goes the message!
My buddies and family are amazing and hanging out together at tavern gigs or meals is always vital. As I see that you can not own a good conversation against the noise I haven't ever been into clubs. I also have two quite adorable and definitely cheeky puppies that are constantly keen to meet fresh individuals.
I try and maintain as physically healthy as possible coming to luke bryan tickets cheap the fitness center many times weekly. I love my athletics where can i buy luke bryan tickets, http://lukebryantickets.asiapak.net, and attempt to perform or watch as numerous a potential. I will often at Hawthorn matches being wintertime. Note: I have noticed the carnage of fumbling matches at stocktake sales, In case that you really considered buying a hobby I really do not brain.
My web-site :: luke bryan tickets luke bryan concert dates 2014 cheap; mouse click the next internet page,