L11n96

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L11n95.gif

L11n95

L11n97.gif

L11n97

Contents

L11n96.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n96 at Knotilus!


Link Presentations

[edit Notes on L11n96's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X7,16,8,17 X9,18,10,19 X17,8,18,9 X22,20,5,19 X20,14,21,13 X14,22,15,21 X15,10,16,11 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, 7, -8, -9, 3, -5, 4, 6, -7, 8, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n96 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^5-2 u v^4+3 u v^3-2 u v^2+2 u v-u-v^5+2 v^4-2 v^3+3 v^2-2 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+3 a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-8 a^3 z^3-5 a^3 z-a^3 z^{-1} +a z^5+3 a z^3+a z (db)
Kauffman polynomial a^9 z^3+3 a^8 z^4-2 a^8 z^2+5 a^7 z^5-6 a^7 z^3+2 a^7 z+a^6 z^8-a^6 z^6+4 a^6 z^4-3 a^6 z^2+a^5 z^9-2 a^5 z^7+5 a^5 z^5-5 a^5 z^3-a^5 z+a^5 z^{-1} +4 a^4 z^8-12 a^4 z^6+11 a^4 z^4-2 a^4 z^2-a^4+a^3 z^9+a^3 z^7-11 a^3 z^5+11 a^3 z^3-4 a^3 z+a^3 z^{-1} +3 a^2 z^8-10 a^2 z^6+7 a^2 z^4+3 a z^7-11 a z^5+9 a z^3-a z+z^6-3 z^4+z^2 (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
4         1-1
2        2 2
0       21 -1
-2      42  2
-4     43   -1
-6    43    1
-8   24     2
-10  34      -1
-12 13       2
-14 2        -2
-161         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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L11n95

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L11n97