L11n182

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L11n181.gif

L11n181

L11n183.gif

L11n183

Contents

L11n182.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n182 at Knotilus!


Link Presentations

[edit Notes on L11n182's Link Presentations]

Planar diagram presentation X8192 X9,21,10,20 X14,5,15,6 X16,8,17,7 X3,10,4,11 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code {1, 11, -5, -10, 3, -9}, {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n182 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^3-3 u^2 v^2+3 u^2 v+2 u v^4-5 u v^3+7 u v^2-5 u v+2 u+3 v^3-3 v^2+v}{u v^2} (db)
Jones polynomial 2 q^{5/2}-5 q^{3/2}+8 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z+a^5 z^{-1} -a^3 z^5-a^3 z^3-2 a z^5-6 a z^3+2 z^3 a^{-1} -7 a z-2 a z^{-1} +3 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-2 a^7 z^3+3 a^6 z^6-5 a^6 z^4+6 a^5 z^7-16 a^5 z^5+16 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-10 a^4 z^6+6 a^4 z^4+a^4 z^2-a^4+2 a^3 z^9+2 a^3 z^7-10 a^3 z^5+10 a^3 z^3-2 a^3 z+8 a^2 z^8-21 a^2 z^6+27 a^2 z^4+3 z^4 a^{-2} -14 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +2 a z^9-3 a z^7+z^7 a^{-1} +10 a z^5+3 z^5 a^{-1} -15 a z^3-7 z^3 a^{-1} +9 a z+3 z a^{-1} -2 a z^{-1} - a^{-1} z^{-1} +3 z^8-8 z^6+19 z^4-20 z^2+5 (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 \
-6-5-4-3-2-10123χ
6         2-2
4        3 3
2       52 -3
0      63  3
-2     76   -1
-4    55    0
-6   47     3
-8  35      -2
-10  4       4
-1213        -2
-141         1
Integral Khovanov Homology

(db, data source)

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

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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