Figure Eight Knot

Author

Eric W. Weisstein
November 9, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/KnotTheory/FigureEightKnot.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FigureEightKnot.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Symbol

Alexander-Briggs Notation

KnotData[knot,"AlexanderBriggsNotation"]//TraditionalForm

Braid Word

KnotData[knot,"BraidWordNotation"]//TraditionalForm

StandardName

KnotData[knot,"StandardName"]

Figure

g1=Graphics[{Thickness[.015],KnotData[knot,"KnotDiagramData"]},ImageSize->200]

Show[KnotData[knot],ImageSize->200]

Graphics3D[{EdgeForm[],ClosedBraid[BraidWord[Link[knot]],Style->TubeStyle][[1]]},ImageSize->200,ViewPoint->{0,0,2},Boxed->False]

Show[ClosedBraid[BraidWord[Link[knot]],Style->TubeStyle],ImageSize->200]

Properties

Classes

KnotData[knot,"Classes"]

KnotClassString[knot]

Properties

props={"AlexanderBriggsList","AlexanderBriggsNotation","AlternateNames","ArfInvariant","BridgeIndex","ColoringNumberSet","ConcordanceOrder","ConwayNotation","ConwayString","ColoringNumberSet","CrossingNumber","ConwayNotation","ConwayString","CrossingNumber","DegreeThreeVassiliev","DegreeTwoVassiliev","Determinant","DowkerList","DowkerNotation","Genus","HyperbolicVolume","NakanishiIndex","Name","OzsvathSzaboTau","SeifertMatrix","Signature","SmoothFourGenus","StandardName","StickNumber","SuperbridgeIndex","ThurstonBennequin","TopologicalFourGenus","UnknottingNumber"};

(#->KnotData[knot,#])&/@props//Column

Polynomials

polys={{"AlexanderPolynomial",x},{"BLMHoPolynomial",x},{"BracketPolynomial",x},{"ConwayPolynomial",x},{"HOMFLYPolynomial",{ℓ,m}},{"JonesPolynomial",t},{"KauffmanPolynomial",{a,z}}};

(#[[1]]->KnotData[knot,#[[1]]][Sequence@@#[[2]]])&/@polys//Column//TraditionalForm

Other Knots with Same Polynomials

Function[poly,{poly,Select[DeleteCases[KnotData[All],knot],KnotData[#,poly]===KnotData[knot,poly]&]}]/@First/@polys//Column

Hyperbolic Volume Complement

Hyperbolic Volume of Complement Sums

V==2 (Sqrt[3]) (∑_(n=1)^∞)(((ψ^((0)))(2 n)-(ψ^((0)))(n))/(n (({{2 n}, {n}}))))

V6

Sum[(1/k),{k,n,2n-1}]//FullSimplify

Timing[a=FullSimplify[2Sqrt[3]Sum[(PolyGamma[0,2 n]-PolyGamma[0,n])(1/(n Binomial[2n,n])),{n,∞}]]]

N[a,20]//Chop

FullSimplify[FunctionExpand[a]]

N[18 I (PolyLog[2,(1/4) (1-I (Sqrt[3]))]-PolyLog[2,(1/4) (1+I (Sqrt[3]))])]

V7

Sum[(1/k),{k,n,2n-1}]

FullSimplify[%]

TimeConstrained[2Sqrt[3]Sum[(PolyGamma[0,2 n]-PolyGamma[0,n])(1/(n Binomial[2n,n])),{n,∞}],60]//Timing

FullSimplify[%]

N[Last[%]]

2Sqrt[3]NSum[(PolyGamma[0,2 n]-PolyGamma[0,n])(1/(n Binomial[2n,n])),{n,∞}]

V

N[%]

V==(∑_(n=1)^∞)((2 sin(((n π)/3)))/(n^2))

V6

FullSimplify[Sum[2 Sin[n Pi/3]/n^2,{n,∞}]]

V7

Sum[2 Sin[n Pi/3]/n^2,{n,∞}]

FullSimplify[%]

V==(1/9) (Sqrt[3]) (∑_(n=1)^∞)((n-(1/2))/((((n-(1/3)))^2) (((n-(2/3)))^2)))

V6

Sqrt[3]/9Sum[((n-(1/2))/((n-1/3)^2(n-2/3)^2)),{n,∞}]

FullSimplify[%]

N[%,20]

V7

Sqrt[3]/9Sum[((n-(1/2))/((n-1/3)^2(n-2/3)^2)),{n,∞}]

FullSimplify[%]

V==2 (∑_(n=0)^∞)(((({{2 n}, {n}})))/((((2 n+1))^2) (16^n)))

V6

2Sum[((Binomial[2n,n])/((((2n+1))^2)16^n)),{n,0,∞}]

FunctionExpand[%]//FullSimplify

N[%,20]

V7

2Sum[((Binomial[2n,n])/((((2n+1))^2)16^n)),{n,0,∞}]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

V==(1/3) (2 π) (-log((π/3))+(∑_(m=1)^∞)((ζ(2 m))/(m (2 m+1) (6^(2 m))))+1)

V6

((2π)/3)(1-Log[Pi/3]+Sum[((Zeta[2m])/(m(2m+1)(6^(2m)))),{m,∞}])

FullSimplify[%]

N[%,20]

((2π)/3)(1-Log[Pi/3]+NSum[((Zeta[2m])/(m(2m+1)(6^(2m)))),{m,∞}])

Factor[FullSimplify[Sum[(((-1))^(k+1))/k,{k,2n+1}]]/.Log[4]->2Log[2]]

V7

((2π)/3)(1-Log[Pi/3]+Sum[((Zeta[2m])/(m(2m+1)(6^(2m)))),{m,∞}])

FullSimplify[%]

V==(Sqrt[3]) (∑_(n=0)^∞)((-(Subscript[H, n])+(Subscript[H, n+1/2])+2 log(2))/(2 ((({{2 n}, {n}})) (2 n+1))))

V6

Sqrt[3]Sum[((1/2) (-HarmonicNumber[n]+HarmonicNumber[(1/2)+n]+2 Log[2]))/(Binomial[2n,n](2n+1)),{n,0,∞}]//FullSimplify

N[%]

Sqrt[3]NSum[((1/2) (-HarmonicNumber[n]+HarmonicNumber[(1/2)+n]+2 Log[2]))/(Binomial[2n,n](2n+1)),{n,0,∞}]

V

N[%]

V7

Sqrt[3]Sum[((1/2) (-HarmonicNumber[n]+HarmonicNumber[(1/2)+n]+2 Log[2]))/(Binomial[2n,n](2n+1)),{n,0,∞}]//FullSimplify

FullSimplify[%]

BBP Formulas for Hyperbolic Volume Complement

<<Utilities`Simplify`

V=(1/9)(Sqrt[3])(3PolyGamma[1,1/3]-2π^2);

V==(Sqrt[3]) (∑_(k=0)^∞)((1/(((3 k+1))^2))-(2/(((3 k+2))^2))+(4/(((3 k+3))^2)))

(Sqrt[3])Sum[(1/(((3k+1))^2))-(2/(((3k+2))^2))+(4/(((3k+3))^2)),{k,0,∞}]

FullSimplify[%]

N[%]

V//N

V==(3/2) (Sqrt[3]) (∑_(k=0)^∞)((1/(((3 k+1))^2))-(1/(((3 k+2))^2)))

3((Sqrt[3])/2)Sum[(1/(((3k+1))^2))-(1/(((3k+2))^2)),{k,0,∞}]

V==(1/2) (3 (Sqrt[3])) (∑_(k=0)^∞)((1/(((3 k+1))^2))-(1/(((3 k+2))^2)))

((3(Sqrt[3]))/2)Sum[(1/(((3k+1))^2))-(1/(((3k+2))^2)),{k,0,∞}]

V==(Sqrt[3]) (∑_(k=0)^∞)((1/(((6 k+1))^2))+(1/(((6 k+2))^2))-(1/(((6 k+4))^2))-(1/(((6 k+5))^2)))

V6

Sqrt[3]Sum[(1/(((6k+1))^2))+(1/(((6k+2))^2))-(1/(((6k+4))^2))-(1/(((6k+5))^2)),{k,0,∞}]

FullSimplify[%]

N[%,20]

V7

Sqrt[3]Sum[(1/(((6k+1))^2))+(1/(((6k+2))^2))-(1/(((6k+4))^2))-(1/(((6k+5))^2)),{k,0,∞}]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

V==(Sqrt[3]) (∑_(k=0)^∞)((2/(((6 k+1))^2))-(3/(((6 k+2))^2))-(1/(((6 k+5))^2)))

(Sqrt[3])Sum[(2/(((6k+1))^2))-(3/(((6k+2))^2))-(1/(((6k+5))^2)),{k,0,∞}]

FullSimplify[%]

N[%]

V==(Sqrt[3]) (∑_(k=0)^∞)((1/(((6 k+1))^2))+(3/(((6 k+4))^2))-(2/(((6 k+5))^2)))

(Sqrt[3])Sum[(1/(((6k+1))^2))+(3/(((6k+4))^2))-(2/(((6k+5))^2)),{k,0,∞}]

FullSimplify[%]

N[%]

V//N

V==(Sqrt[3]) (∑_(k=0)^∞)(((-1))^k) ((1/(((3 k+1))^2))+(1/(((3 k+2))^2)))

(Sqrt[3])Sum[(((-1))^k)((1/(((3k+1))^2))+(1/(((3k+2))^2))),{k,0,∞}]

FullSimplify[%]

N[%]

V==(1/243) (2 (Sqrt[3])) (∑_(m=0)^∞)(1/(729^m))((243/(((12 m+1))^2))-(243/(((12 m+2))^2))-(324/(((12 m+3))^2))-(81/(((12 m+4))^2))+(27/(((12 m+5))^2))-(9/(((12 m+7))^2))+(9/(((12 m+8))^2))+(12/(((12 m+9))^2))+(3/(((12 m+10))^2))-(1/(((12 m+11))^2)))

((2(Sqrt[3]))/243) Sum[(1/(729^m))((243/(((1+12 m))^2))-(243/(((2+12 m))^2))-(324/(((3+12 m))^2))-(81/(((4+12 m))^2))+(27/(((5+12 m))^2))-(9/(((7+12 m))^2))+(9/(((8+12 m))^2))+(12/(((9+12 m))^2))+(3/(((10+12 m))^2))-(1/(((11+12 m))^2))),{m,0,∞}]

N[%]

V//N

V==(1/177147)(2 (Sqrt[3])) (∑_(m=0)^∞)(1/(531441^m))((177147/(((24 m+1))^2))-(177147/(((24 m+2))^2))-(236196/(((24 m+3))^2))-(59049/(((24 m+4))^2))+(19683/(((24 m+5))^2))-(6561/(((24 m+7))^2))+(6561/(((24 m+8))^2))+(8748/(((24 m+9))^2))+(2187/(((24 m+10))^2))-(729/(((24 m+11))^2))+(243/(((24 m+13))^2))-(243/(((24 m+14))^2))-(324/(((24 m+15))^2))-(81/(((24 m+16))^2))+(27/(((24 m+17))^2))-(9/(((24 m+19))^2))+(9/(((24 m+20))^2))+(12/(((24 m+21))^2))+(3/(((24 m+22))^2))-(1/(((24 m+23))^2)))

((2(Sqrt[3]))/177147)Sum[(1/(531441^m))((177147/(((1+24 m))^2))-(177147/(((2+24 m))^2))-(236196/(((3+24 m))^2))-(59049/(((4+24 m))^2))+(19683/(((5+24 m))^2))-(6561/(((7+24 m))^2))+(6561/(((8+24 m))^2))+(8748/(((9+24 m))^2))+(2187/(((10+24 m))^2))-(729/(((11+24 m))^2))+(243/(((13+24 m))^2))-(243/(((14+24 m))^2))-(324/(((15+24 m))^2))-(81/(((16+24 m))^2))+(27/(((17+24 m))^2))-(9/(((19+24 m))^2))+(9/(((20+24 m))^2))+(12/(((21+24 m))^2))+(3/(((22+24 m))^2))-(1/(((23+24 m))^2))),{m,0,∞}]

FullSimplify[%]

N[%]

V==(Sqrt[3]) (∑_(k=0)^∞)(((-1))^k) ((1/(((9 k+1))^2))+(1/(((9 k+2))^2))-(1/(((9 k+4))^2))-(1/(((9 k+5))^2))+(1/(((9 k+7))^2))+(1/(((9 k+8))^2)))

(Sqrt[3])Sum[(((-1))^k)((1/(((9k+1))^2))+(1/(((9k+2))^2))-(1/(((9k+4))^2))-(1/(((9k+5))^2))+(1/(((9k+7))^2))+(1/(((9k+8))^2))),{k,0,∞}]

FullSimplify[%]

N[%]

V//N

V==(2/9) (Sqrt[3]) (∑_(n=0)^∞)((((-1))^n)/(27^n)) ((9/(((6 n+1))^2))-(9/(((6 n+2))^2))-(12/(((6 n+3))^2))-(3/(((6 n+4))^2))+(1/(((6 n+5))^2)))

V6

Timing[b=FullSimplify[((Sqrt[3])/9) Sum[((((-1))^n)/(27^n))((18/(((6n+1))^2))-(18/(((6n+2))^2))-(24/(((6n+3))^2))-(6/(((6n+4))^2))+(2/(((6n+5))^2))),{n,0,∞}]]]

N[b,20]

V7

((Sqrt[3])/9) Sum[((((-1))^n)/(27^n))((18/(((6n+1))^2))-(18/(((6n+2))^2))-(24/(((6n+3))^2))-(6/(((6n+4))^2))+(2/(((6n+5))^2))),{n,0,∞}]

FullSimplify[%]

N[%]

V//N

Search

Last/@(BasisSum[#,k]&/@FindBasis[(V/(Sqrt[3])),-27,6,BasisPower->2,WorkingPrecision->300])//ColumnForm//TraditionalForm

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,1}],{-3^k,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,1}],{-2^k,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,0,0},{n,64},{p,1}],{-1,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{2^k,n,p}]

Monitor[Do[If[(b=FindBasis[(V/(Sqrt[3])),3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[(Last/@(BasisSum[#,m]&/@b))]],{k,16},{n,64},{p,2,2}],{3^k,n,p}]

Hyperbolic Volume of Complement Integrals

V==-2 (∫_0^(π/3))log(2 sin((t/2)))dt

FullSimplify[-2Integrate[Log[2Sin[t/2]],{t,0,Pi/3}]]

FullSimplify[FunctionExpand[%]]

V==-2 (∫_0^1)((log(y))/(Sqrt[1-((((y/2)))^2)]))dy

-2Integrate[((Log[y])/(Sqrt[1-(y/2)^2])),{y,0,1}]

V==-(Sqrt[3]) (∫_0^1)((log(y))/((y^2)-y+1))dy

-Sqrt[3]Integrate[((Log[y])/(1-y+y^2)),{y,0,1}]

-Sqrt[3]Integrate[((Log[y])/(1-y+y^2)),{y,0,1}]

FullSimplify[%]

V==2 (Sqrt[3])(∫_0^(1/2))(((s+1) log(s+1)-(1-s) log(1-s))/((1-(s^2)) (Sqrt[1-4 (s^2)])))ds

2Sqrt[3]Integrate[(((1+s)Log[1+s]-(1-s)Log[1-s])/((1-s^2)Sqrt[1-4s^2])),{s,0,1/2}]//Timing

2Sqrt[3]NIntegrate[(((1+s)Log[1+s]-(1-s)Log[1-s])/((1-s^2)Sqrt[1-4s^2])),{s,0,1/2}]

2(Sqrt[3])NIntegrate[((Log[((1+s)/(1-s))])/((1-(s^2))(Sqrt[1-4(s^2)]))),{s,0,1/2}]-2(Sqrt[3])NIntegrate[((-s Log[1-(s^2)])/((1-(s^2))(Sqrt[1-4(s^2)]))),{s,0,1/2}]

Hyperbolic Volume of Complement Clausen

V==2 (Subscript[Cl, 2])( (π/3))==3 (Subscript[Cl, 2])( ((2 π)/3))

N[{2ClausenCl[2,Pi/3],3ClausenCl[2,2Pi/3]},20]

FullSimplify[{2ClausenCl[2,Pi/3],3ClausenCl[2,2Pi/3]}]

V==2 Im((Subscript[Li, 2])((e^((i π)/3))))

N[2Im[PolyLog[2,Exp[I Pi/3]]],20]

N[I(PolyLog[2,Exp[-I π/3]]-PolyLog[2,Exp[I π/3]]),20]//Chop