L11n214

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L11n213.gif

L11n213

L11n215.gif

L11n215

Contents

L11n214.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n214's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,13,21,14 X5,14,6,15 X7,16,8,17 X15,6,16,7 X4,20,5,19
Gauss code {1, -2, 3, -11, -8, 10, -9, 6}, {-4, -1, 2, -3, 7, 8, -10, 9, -5, 4, 11, -7, -6, 5}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L11n214 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-3 t(2)^3 t(1)^2+5 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-4 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+5 a^7 z^5+10 a^7 z^3+10 a^7 z+3 a^7 z^{-1} -2 a^5 z^5-8 a^5 z^3-9 a^5 z-2 a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+a^{13} z+5 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2+5 a^{11} z^7-5 a^{11} z^5+a^{11} z^3+a^{11} z+3 a^{10} z^8-3 a^{10} z^4+a^{10}+a^9 z^9+3 a^9 z^7-2 a^9 z^5-6 a^9 z^3+4 a^9 z-a^9 z^{-1} +4 a^8 z^8-7 a^8 z^6+8 a^8 z^4-11 a^8 z^2+3 a^8+a^7 z^9-2 a^7 z^7+9 a^7 z^5-20 a^7 z^3+14 a^7 z-3 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+4 a^6 z^4-7 a^6 z^2+3 a^6+3 a^5 z^5-10 a^5 z^3+10 a^5 z-2 a^5 z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-10χ
-4         22
-6        21-1
-8       51 4
-10      42  -2
-12     65   1
-14    55    0
-16   45     -1
-18  25      3
-20 14       -3
-22 2        2
-241         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L11n213.gif

L11n213

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L11n215