Knot
Invariants| Knot
Invariants |
In General > s.a. energy;
topological field theories.
* Idea: Can be generated
as coefficients of polynomials using group theory; Braids can be used as a
calculational tool.
* Alexander polynomial:
Satisfies 
k+1(t)
= k(t) 1(t).
Examples: unknot(t)
= 1; trefoil(t)
= t 2 – t + 1; figure
8(t)
= t 2 – 3t + 1.
* Conway polynomial:
Defined by Cunknot(z)
= 1 and the skein relations CL_+(z) – CL_–(z)
= z CL_0(z),
where the L's
refer to the two possibilities
for a crossing in a plane projection and to the crossing
replaced
by the lines reconnected in such a way that they do not cross; To extend it
to double intersections, define CL_W
= CL_0 and CL_I
= 
(CL_+ +
CL_–) for the two
routings.
* Helicity: A second order integral in field amplitudes.
* Kauffman bracket polynomial:
* Unknotting number: The minimal number of self-crossings needed to
obtain the unknot.
* And quantum groups:
For every simple Lie algebra 
there
is a Hopf algebra Uq(),
and a polynomial link invariant; For example,
Uq(sl2)
corresponds to the Jones polynomial; HOMFLY.
References > s.a. knot theory [for generalized knots].
@ General: Buck & Flapan JPA(07)
[topological characterization].
@ Polynomials: Akutsu & Wadati JPSJ(87),
et al JPSJ(87), JPSJ(88);
Jones AM(87); Lickorish & Millett
Top(87);
Deguchi et al JPSJ(88);
Akutsu et al JPSJ(88);
Kauffman IJMPA(90);
Broda PLB(91),
JMP(94); Suffczynski
PLA(96)
[rep]; Labastida & Mariño JKTR(02)m.QA/01.
@ Alexander-Conway polynomial: Kauffman Top(81);
Friedman Top(04)
[generalization]; Tsutsumi & Yamada Top(04) [and Dehn surgery]; Garoufalidis & Teichner
JDG(04) [trivial polynomial].
@ Jones polynomial: Jones BAMS(85); Kauffman Top(87);
Witten CMP(89)
[and Chern-Simons theory]; Zulli Top(95),
Chang & Shrock
PhyA(01)mp/01 [computation];
Subramaniam & Ramadevi qp/02,
Lomonaco & Kauffman qp/06,
Garnerone et al LP(06)qp [quantum
computation]; Loebl a0705 [from
the permanent of a matrix]; Kauffman & Lomonaco a0706-in
[algorithms]; > s.a. spin networks.
@ HOMFLY: Freyd et al BAMS(85);
Labastida & Mariño IJMPA(95)
[torus knots, from Chern-Simons]; Labastida & Pérez JMP(96)qa/95 [for
torus knots, and Kauffman].
@ Third-order invariant: Berger JPA(90);
Evans & Berger in(92).
@ Enumeration: Jacobsen & Zinn-Justin mp/01,
mp/01 [transfer matrix approach].
@ Classification algorithms: Aneziris ht/94-in, qa/95, qa/95, qa/96, qa/96-in, qa/97-in;
Flammini & Stasiak PRS(07).
@ And quantum groups: Turaev 94 [r Kuperberg BAMS(96)]; Sawin BAMS(96);
Nikshych et al T&A(02).
@ And 3-manifolds: Blanchet et al Top(92)
[Kauffman bracket and 3-manifold invariants]; Kauffman & Baadhio ed-93 [quantum
field theory methods, quantum topology]; Eisermann Top(04).
@ And embedded graphs: Moffatt EJC(08)
[Bollobás–Riordan polynomial].
@ Related topics: Adams PAMS(89) + refs
[Gromov invariant]; O'Hara Top(91)
[energy]; Akhmetiev & Ruzmaikin
JGP(95)
[Sato-Levine as 4th-order integral]; Álvarez & Labastida JKTR(96)
[torus knots, Vassiliev]; Gukov a0706-in
[homological, from topological gauge theories].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008