L11n281

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L11n280.gif

L11n280

L11n282.gif

L11n282

Contents

L11n281.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n281 at Knotilus!


Link Presentations

[edit Notes on L11n281's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X22,17,11,18 X20,11,21,12 X16,21,17,22 X4,15,1,16 X19,10,20,5
Gauss code {1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {8, 2, 4, -5, 10, -9, 7, 6, -11, -8, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n281 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial 2 q^{-2} -3 q^{-3} +8 q^{-4} -7 q^{-5} +9 q^{-6} -8 q^{-7} +6 q^{-8} -4 q^{-9} + q^{-10} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} +a^8 z^4+2 a^8 z^2+4 a^8 z^{-2} +4 a^8-a^6 z^6-4 a^6 z^4-8 a^6 z^2-5 a^6 z^{-2} -10 a^6+2 a^4 z^4+6 a^4 z^2+2 a^4 z^{-2} +6 a^4 (db)
Kauffman polynomial z^4 a^{12}+4 z^5 a^{11}-4 z^3 a^{11}+a^{11} z^{-1} +6 z^6 a^{10}-8 z^4 a^{10}+2 z^2 a^{10}-a^{10} z^{-2} +4 z^7 a^9-z^5 a^9-3 z^3 a^9-5 z a^9+5 a^9 z^{-1} +z^8 a^8+7 z^6 a^8-11 z^4 a^8+z^2 a^8-4 a^8 z^{-2} +5 a^8+5 z^7 a^7-6 z^5 a^7+7 z^3 a^7-15 z a^7+9 a^7 z^{-1} +z^8 a^6+z^6 a^6+z^4 a^6-8 z^2 a^6-5 a^6 z^{-2} +10 a^6+z^7 a^5-z^5 a^5+6 z^3 a^5-10 z a^5+5 a^5 z^{-1} +3 z^4 a^4-7 z^2 a^4-2 a^4 z^{-2} +6 a^4 (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 \
-8-7-6-5-4-3-2-10χ
-3        22
-5       32-1
-7      5  5
-9     23  1
-11    75   2
-13   34    1
-15  35     -2
-17 13      2
-19 3       -3
-211        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-8 {\mathbb Z}
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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{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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L11n280

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L11n282