L11n40

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L11n39.gif

L11n39

L11n41.gif

L11n41

Contents

L11n40.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n40 at Knotilus!


Link Presentations

[edit Notes on L11n40's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,20,16,21 X17,22,18,5 X21,16,22,17 X19,10,20,11 X11,2,12,3
Gauss code {1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n40 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^5-3 u v^4+3 u v^3-2 u v^2+u v+v^4-2 v^3+3 v^2-3 v+2}{\sqrt{u} v^{5/2}} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{2}{q^{25/2}}-\frac{4}{q^{23/2}}+\frac{6}{q^{21/2}}-\frac{7}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^7 a^9-4 z^5 a^9-2 z^3 a^9+4 z a^9+2 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-12 z^3 a^7-9 z a^7-2 a^7 z^{-1} (db)
Kauffman polynomial -3 z^2 a^{16}+a^{16}-z^5 a^{15}-3 z^3 a^{15}+z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-2 z^2 a^{14}-4 z^7 a^{13}+8 z^5 a^{13}-4 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -3 z^8 a^{12}+7 z^6 a^{12}-7 z^4 a^{12}+9 z^2 a^{12}-3 a^{12}-z^9 a^{11}-z^7 a^{11}+7 z^5 a^{11}+z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+13 z^6 a^{10}-9 z^4 a^{10}+z^2 a^{10}-z^9 a^9+2 z^7 a^9+4 z^5 a^9-10 z^3 a^9+7 z a^9-2 a^9 z^{-1} -z^8 a^8+3 z^6 a^8+z^4 a^8-7 z^2 a^8+3 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+9 z a^7-2 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       3  3
-12      21  -1
-14     53   2
-16    33    0
-18   44     0
-20  23      1
-22 24       -2
-24 2        2
-262         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11n39.gif

L11n39

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L11n41