L11n360

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L11n359.gif

L11n359

L11n361.gif

L11n361

Contents

L11n360.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n360 at Knotilus!


Link Presentations

[edit Notes on L11n360's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X17,14,18,15 X15,22,16,17 X21,16,22,5 X2,12,3,11
Gauss code {1, -11, 5, -3}, {-8, 4, -7, 6, -10, 9}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 8, -9, 10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11n360 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(3)-1)^2}{\sqrt{t(1)} t(3)} (db)
Jones polynomial q^3-2 q^2+3 q-3+4 q^{-1} -2 q^{-2} +3 q^{-3} + q^{-6} - q^{-7} (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^2 a^6-2 a^6+z^4 a^4+5 z^2 a^4+a^4 z^{-2} +5 a^4-z^4 a^2-3 z^2 a^2-2 a^2 z^{-2} -4 a^2-z^4-2 z^2+ z^{-2} +z^2 a^{-2} + a^{-2} (db)
Kauffman polynomial a^7 z^7-6 a^7 z^5+9 a^7 z^3-3 a^7 z+a^6 z^8-7 a^6 z^6+14 a^6 z^4-12 a^6 z^2+5 a^6+a^5 z^7-9 a^5 z^5+18 a^5 z^3-10 a^5 z+2 a^4 z^8-16 a^4 z^6+39 a^4 z^4-38 a^4 z^2-a^4 z^{-2} +15 a^4+a^3 z^9-5 a^3 z^7+2 a^3 z^5+13 a^3 z^3-13 a^3 z+2 a^3 z^{-1} +3 a^2 z^8-18 a^2 z^6+z^6 a^{-2} +35 a^2 z^4-4 z^4 a^{-2} -31 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2- a^{-2} +a z^9-3 a z^7+2 z^7 a^{-1} -3 a z^5-8 z^5 a^{-1} +10 a z^3+6 z^3 a^{-1} -7 a z-z a^{-1} +2 a z^{-1} +2 z^8-8 z^6+6 z^4-2 z^2- z^{-2} +3 (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
7           11
5          1 -1
3         21 1
1       121  0
-1      142   1
-3     123    2
-5    142     1
-7   113      3
-9   11       0
-11 111        1
-13            0
-151           -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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L11n359

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L11n361