L11a82

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L11a81.gif

L11a81

L11a83.gif

L11a83

Contents

L11a82.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a82 at Knotilus!


Link Presentations

[edit Notes on L11a82's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,8,17,7 X22,13,5,14 X14,17,15,18 X20,10,21,9 X18,21,19,22 X8,16,9,15 X10,20,11,19 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -8, 6, -9, 11, -2, 4, -5, 8, -3, 5, -7, 9, -6, 7, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
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BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
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A Morse Link Presentation L11a82 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^4-10 u v^3+14 u v^2-6 u v+u+v^5-6 v^4+14 v^3-10 v^2+2 v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+3 q^{5/2}-\frac{22}{q^{5/2}}-8 q^{3/2}+\frac{21}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+14 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+3 z^3 a^5+3 z a^5+2 a^5 z^{-1} -2 z^5 a^3-4 z^3 a^3-8 z a^3-4 a^3 z^{-1} -z^5 a+3 z^3 a+6 z a+3 a z^{-1} +2 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} -z a^{-3} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-10 a^7 z^5+8 a^7 z^3-3 a^7 z+6 a^6 z^8-12 a^6 z^6+5 a^6 z^4-a^6 z^2+a^6+4 a^5 z^9+5 a^5 z^7-34 a^5 z^5+36 a^5 z^3-15 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+17 a^4 z^8-47 a^4 z^6+40 a^4 z^4-14 a^4 z^2+2 a^4+9 a^3 z^9-a^3 z^7-40 a^3 z^5+z^5 a^{-3} +53 a^3 z^3-2 z^3 a^{-3} -24 a^3 z+z a^{-3} +4 a^3 z^{-1} +a^2 z^{10}+19 a^2 z^8-51 a^2 z^6+3 z^6 a^{-2} +53 a^2 z^4-4 z^4 a^{-2} -24 a^2 z^2+z^2 a^{-2} +3 a^2+5 a z^9+4 a z^7+6 z^7 a^{-1} -26 a z^5-9 z^5 a^{-1} +34 a z^3+7 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +8 z^8-14 z^6+16 z^4-11 z^2+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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         61 5
2        82  -6
0       106   4
-2      129    -3
-4     109     1
-6    812      4
-8   610       -4
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}_2 {\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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L11a81.gif

L11a81

L11a83.gif

L11a83