L11n131

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L11n130.gif

L11n130

L11n132.gif

L11n132

Contents

L11n131.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n131's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X16,10,17,9 X20,14,21,13 X22,16,7,15 X19,4,20,5 X14,22,15,21
Gauss code {1, 4, -3, 10, -5, -6}, {6, -1, 7, 3, -2, 5, 8, -11, 9, -7, -4, 2, -10, -8, 11, -9}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n131 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+2 t(1) t(2)-2 t(2)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{15/2}-2 q^{13/2}+4 q^{11/2}-5 q^{9/2}+6 q^{7/2}-7 q^{5/2}+5 q^{3/2}-5 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-3} +z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} +4 z^3 a^{-1} -13 z^3 a^{-3} +4 z^3 a^{-5} +6 z a^{-1} -12 z a^{-3} +5 z a^{-5} +3 a^{-1} z^{-1} -5 a^{-3} z^{-1} +2 a^{-5} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -4 z^4 a^{-8} +4 z^2 a^{-8} - a^{-8} +2 z^7 a^{-7} -7 z^5 a^{-7} +5 z^3 a^{-7} +2 z^8 a^{-6} -6 z^6 a^{-6} +3 z^4 a^{-6} +z^9 a^{-5} -2 z^7 a^{-5} +2 z^5 a^{-5} -7 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +3 z^8 a^{-4} -10 z^6 a^{-4} +15 z^4 a^{-4} -14 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} -4 z^7 a^{-3} +13 z^5 a^{-3} -20 z^3 a^{-3} +15 z a^{-3} -5 a^{-3} z^{-1} +z^8 a^{-2} -3 z^6 a^{-2} +10 z^4 a^{-2} -11 z^2 a^{-2} +5 a^{-2} +4 z^5 a^{-1} +a z^3-7 z^3 a^{-1} -a z+8 z a^{-1} -3 a^{-1} z^{-1} +2 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 \
-2-101234567χ
16         1-1
14        1 1
12       31 -2
10      21  1
8     43   -1
6    32    1
4   24     2
2  33      0
0  3       3
-212        -1
-41         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-2 {\mathbb Z} {\mathbb Z}
r=-1 {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\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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L11n130

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L11n132