L11n20

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L11n19.gif

L11n19

L11n21.gif

L11n21

Contents

L11n20.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n20 at Knotilus!


Link Presentations

[edit Notes on L11n20's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X17,1,18,4 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X9,18,10,19 X11,20,12,21 X19,10,20,11 X2,16,3,15
Gauss code {1, -11, -5, 3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n20 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)-3 a^9 z-2 a^9 z^{-1} +a^7 z^5+4 a^7 z^3+7 a^7 z+4 a^7 z^{-1} +a^5 z^5+2 a^5 z^3-a^5 z^{-1} -2 a^3 z^3-4 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial a^{12} z^6-5 a^{12} z^4+7 a^{12} z^2-2 a^{12}+a^{11} z^7-3 a^{11} z^5+a^{11} z^3+a^{11} z+a^{10} z^8-2 a^{10} z^6-a^{10} z^4+2 a^{10} z^2-a^{10}+a^9 z^9-4 a^9 z^7+11 a^9 z^5-17 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+21 a^8 z^4-21 a^8 z^2+6 a^8+a^7 z^9-3 a^7 z^7+10 a^7 z^5-16 a^7 z^3+13 a^7 z-4 a^7 z^{-1} +2 a^6 z^8-7 a^6 z^6+17 a^6 z^4-15 a^6 z^2+5 a^6+2 a^5 z^7-4 a^5 z^5+5 a^5 z^3+a^5 z-a^5 z^{-1} +a^4 z^6+a^4 z^2-a^4+3 a^3 z^3-5 a^3 z+a^3 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
-2         22
-4        32-1
-6       3  3
-8      33  0
-10     43   1
-12    23    1
-14   34     -1
-16   2      2
-18 13       -2
-20          0
-221         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\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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L11n19

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L11n21