L11n65

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L11n64.gif

L11n64

L11n66.gif

L11n66

Contents

L11n65.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n65's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X17,20,18,21 X11,18,12,19 X19,12,20,13 X15,22,16,5 X21,16,22,17 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -7, 8, -4, 5, -6, 4, -8, 7}
A Braid Representative
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A Morse Link Presentation L11n65 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^3+t(2)^3-2 t(1) t(2)^2-2 t(2)+t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{3}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +2 a^{11} z+2 a^{11} z^{-1} -a^9 z^{-1} -a^7 z^5-3 a^7 z^3+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-4 a^5 z-a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^8-7 a^{14} z^6+15 a^{14} z^4-12 a^{14} z^2+3 a^{14}+a^{13} z^9-7 a^{13} z^7+14 a^{13} z^5-10 a^{13} z^3+4 a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-16 a^{12} z^6+37 a^{12} z^4-29 a^{12} z^2+7 a^{12}+a^{11} z^9-8 a^{11} z^7+19 a^{11} z^5-17 a^{11} z^3+9 a^{11} z-2 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+20 a^{10} z^4-16 a^{10} z^2+4 a^{10}+2 a^9 z^5-4 a^9 z^3+3 a^9 z-a^9 z^{-1} +2 a^8 z^6-4 a^8 z^4+a^7 z^7-2 a^7 z^5-a^7 z^3+2 a^7 z-a^7 z^{-1} +a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          110
-8         2  2
-10        11  0
-12      132   0
-14      12    1
-16    132     0
-18   1 1      2
-20   12       -1
-22 11         0
-24            0
-261           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2} {\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n64

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L11n66