L11n42

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L11n41.gif

L11n41

L11n43.gif

L11n43

Contents

L11n42.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n42's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X9,18,10,19 X3849 X5,14,6,15 X15,22,16,5 X17,20,18,21 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, 8, -9, 7}
A Braid Representative
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A Morse Link Presentation L11n42 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(1) t(2)^3-4 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-4 t(2)+3}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-3 z^3 a^9-2 z a^9-a^9 z^{-1} -z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1} (db)
Kauffman polynomial 3 a^{14} z^4-8 a^{14} z^2+3 a^{14}+a^{13} z^7-a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-10 a^{12} z^6+24 a^{12} z^4-24 a^{12} z^2+7 a^{12}+a^{11} z^9-4 a^{11} z^7+10 a^{11} z^5-11 a^{11} z^3+8 a^{11} z-2 a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+25 a^{10} z^4-16 a^{10} z^2+4 a^{10}+a^9 z^9-4 a^9 z^7+9 a^9 z^5-7 a^9 z^3+4 a^9 z-a^9 z^{-1} +a^8 z^8-2 a^8 z^6+2 a^8 z^4-a^8 z^2+a^7 z^7-a^7 z^5-3 a^7 z^3+3 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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        110
-8       2  2
-10      21  -1
-12     42   2
-14    33    0
-16   33     0
-18  13      2
-20 23       -1
-22 1        1
-242         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n41

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L11n43