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 2t + 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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