L11n19

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L11n18.gif

L11n18

L11n20.gif

L11n20

Contents

L11n19.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n19 at Knotilus!


Link Presentations

[edit Notes on L11n19's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 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 X15,2,16,3
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.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.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 L11n19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^5-u v^4-v+2}{\sqrt{u} v^{5/2}} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z a^{13}-2 a^{13} z^{-1} +z^5 a^{11}+6 z^3 a^{11}+9 z a^{11}+4 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-5 z a^9-a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7-a^7 z^{-1} (db)
Kauffman polynomial -z^2 a^{16}+2 a^{16}-z a^{15}-z^2 a^{14}+a^{14}-z^5 a^{13}+5 z^3 a^{13}-7 z a^{13}+2 a^{13} z^{-1} -z^8 a^{12}+7 z^6 a^{12}-17 z^4 a^{12}+18 z^2 a^{12}-6 a^{12}-z^9 a^{11}+7 z^7 a^{11}-18 z^5 a^{11}+24 z^3 a^{11}-15 z a^{11}+4 a^{11} z^{-1} -2 z^8 a^{10}+12 z^6 a^{10}-22 z^4 a^{10}+17 z^2 a^{10}-5 a^{10}-z^9 a^9+6 z^7 a^9-11 z^5 a^9+9 z^3 a^9-4 z a^9+a^9 z^{-1} -z^8 a^8+5 z^6 a^8-5 z^4 a^8-z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+5 z a^7-a^7 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
-6         11
-8        110
-10       1  1
-12     111  1
-14    131   1
-16   111    1
-18   32     1
-20 111      1
-22 12       -1
-241         -1
-261         -1
Integral Khovanov Homology

(db, data source)

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

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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L11n18

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