L11n109

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L11n108.gif

L11n108

L11n110.gif

L11n110

Contents

L11n109.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n109 at Knotilus!


Link Presentations

[edit Notes on L11n109's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X7,18,8,19 X19,22,20,5 X13,20,14,21 X21,14,22,15 X9,16,10,17 X15,10,16,11 X17,8,18,9 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, -7, 8, 11, -2, -5, 6, -8, 7, -9, 3, -4, 5, -6, 4}
A Braid Representative
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A Morse Link Presentation L11n109 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-u v^2-u v+u+v^3-v^2-v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{21/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{25/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +2 z a^{11}+2 a^{11} z^{-1} +z a^9-a^9 z^{-1} -z^5 a^7-4 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-3 z a^5-a^5 z^{-1} (db)
Kauffman polynomial -z^8 a^{14}+7 z^6 a^{14}-15 z^4 a^{14}+11 z^2 a^{14}-3 a^{14}-z^9 a^{13}+7 z^7 a^{13}-15 z^5 a^{13}+12 z^3 a^{13}-4 z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+15 z^6 a^{12}-34 z^4 a^{12}+28 z^2 a^{12}-7 a^{12}-z^9 a^{11}+8 z^7 a^{11}-20 z^5 a^{11}+21 z^3 a^{11}-10 z a^{11}+2 a^{11} z^{-1} -z^8 a^{10}+8 z^6 a^{10}-18 z^4 a^{10}+14 z^2 a^{10}-4 a^{10}+2 z^3 a^9-4 z a^9+a^9 z^{-1} -z^6 a^8+4 z^4 a^8-3 z^2 a^8-z^7 a^7+4 z^5 a^7-3 z^3 a^7-z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+4 z^3 a^5-3 z a^5+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         1  1
-10       111  1
-12      131   1
-14       1    1
-16    122     -1
-18   1        1
-20   11       0
-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} {\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{2} {\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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L11n108.gif

L11n108

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L11n110