Knot Invariants| Knot Invariants |
In General > s.a. Braids; knot theory
[for generalized knots]; types of homology.
* Idea: They can be generated
as coefficients of polynomials using group theory; Braids can be used as a
calculational tool.
@ General references: Buck & Flapan JPA(07) [topological characterization];
Chmutov et al 12 [specially Vassiliev invariants];
Li 15.
@ Enumeration: Jacobsen & Zinn-Justin mp/01,
mp/01 [transfer-matrix approach].
@ Classification algorithms: Aneziris ht/94-conf,
qa/95,
qa/95,
qa/96,
qa/96-conf,
qa/97-proc;
Flammini & Stasiak PRS(07).
@ 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;
Zodinmawia & Ramadevi a1209 [for non-torus knots and links];
Dolotin & Morozov NPB(14)-a1308 [tensor-algebra approach];
Witten a1401 [gauge-theory approach].
@ Space of knot invariants: Arthamonov et al TMP(14) [differential hierarchy of knot polynomials].
> Online resources:
see MathWorld page.
Specific Invariants > s.a. energy.
* Alexander polynomial:
Satisfies Δk+1(t)
= Δk(t)
Δ1(t);
Examples: Δunknot(t)
= 1; Δtrefoil(t)
= t 2 − t + 1;
Δfigure 8(t)
= t 2 − 3t + 1.
* Conway polynomial:
Defined by Cunknot(z)
= 1 and the skein relations CL+(z)
− CL−(z)
= z CL0(z),
where the Ls 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 CLW
= CL0 and
CLI
= \(1\over2\)(CL+
+ CL−)
for the two possible routings.
* Helicity: A second-order integral in field amplitudes.
* HOMFLY polynomial: A 2-variable oriented knot polynomial
PL(a,z) motivated by the
Jones polynomial, and named after its co-discoverers Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter;
> s.a. MathWorld page;
Wikipedia page.
* Kauffman bracket polynomial:
* Unknotting number:
The minimal number of self-crossings needed to obtain the unknot.
@ Alexander-Conway polynomial:
Kauffman Top(81);
Friedman Top(04) [generalization];
Tsutsumi & Yamada Top(04) [and Dehn surgery];
Garoufalidis & Teichner JDG(04) [trivial polynomial];
Kashaev a2007.
@ 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 SPIE(06)qp,
Garnerone et al LP(06)qp [quantum computation];
Loebl & Moffatt AAM(11)-a0705 [from the permanent of a matrix];
Kauffman & Lomonaco SPIE(07)-a0706 [algorithms];
Gelca a0901 [and quantum mechanics];
Kuperberg ToC-a0908 [approximation];
Gaiotto & Witten ATMP(12)-a1106 [from 4D gauge theory];
Allen & Swenberg a2011 [and causality];
> s.a. spin networks.
@ Kauffman invariant:
Astorino PRD(10)-a1005,
Liu AP(10) [and Chern-Simons theory].
@ HOMFLY polynomial:
Freyd et al BAMS(85);
Mironov et al TMP(13) [genus expansion];
Anokhina & Morozov TMP(14) [cabling procedure].
@ HOMFLY polynomial, other types:
Labastida & Mariño IJMPA(95) [torus knots, from CS theory];
Morozov et al PLB(14)-a1407,
PLB(16)-a1511 [virtual knots].
@ Vassiliev:
Chmutov et al 12 [r BAMS(13)].
@ Third-order invariant: Berger JPA(90);
Evans & Berger in(92).
Related Topics
> s.a. Link Theory; topological field theories.
* And quantum groups:
For every simple Lie algebra \(\cal G\) there is a Hopf algebra \(U_q\)(\(\cal G\)),
and a polynomial link invariant; For example, \(U_q\)(sl2)
corresponds to the Jones polynomial; HOMFLY.
* And topological field theories:
The expectation value of Wilson loop operators in three-dimensional SO(N)
Chern-Simons gauge theory gives a known knot invariant, the Kauffman polynomial.
@ For torus knots: Labastida & Pérez JMP(96)qa/95 [HOMFLY and Kauffman];
Álvarez & Labastida JKTR(96) [Vassiliev];
Stevan AHP(10)-a1003 [HOMFLY and Kauffman invariants, from Chern-Simons theory].
@ And quantum groups: Sawin BAMS(96)qa/95;
Nikshych et al T&A(02);
Turaev 10.
@ 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);
Grishanov & Vassiliev T&A(08) [non-trivial, weight systems].
@ 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];
Gukov a0706-conf
[homological, from topological gauge theories];
Morozov TMP(16)-a1509 [p-adic knot invariants];
Pavlyuk UJP-a1511 [holographic principle];
Elliot & Gukov a1505 [hyperpolynomials].
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