L11n378

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L11n377.gif

L11n377

L11n379.gif

L11n379

Contents

L11n378.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n378 at Knotilus!


Link Presentations

[edit Notes on L11n378's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X22,18,19,17 X11,20,12,21 X19,12,20,13 X18,22,5,21 X9,16,10,17 X2,14,3,13
Gauss code {1, -11, -5, 3}, {-8, 7, 9, -6}, {-4, -1, 2, 5, -10, 4, -7, 8, 11, -2, -3, 10, 6, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n378 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} (db)
Jones polynomial - q^{-7} + q^{-6} -2 q^{-5} +3 q^{-4} -2 q^{-3} +3 q^{-2} - q^{-1} +3 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6 z^{-2} -2 a^6+z^4 a^4+4 z^2 a^4+4 a^4 z^{-2} +7 a^4-3 z^2 a^2-5 a^2 z^{-2} -8 a^2+2 z^{-2} +3 (db)
Kauffman polynomial a^7 z^7-6 a^7 z^5+11 a^7 z^3-6 a^7 z+a^7 z^{-1} +a^6 z^8-5 a^6 z^6+7 a^6 z^4-4 a^6 z^2-a^6 z^{-2} +3 a^6+3 a^5 z^7-16 a^5 z^5+27 a^5 z^3-19 a^5 z+5 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+6 a^4 z^4-11 a^4 z^2-4 a^4 z^{-2} +10 a^4+2 a^3 z^7-10 a^3 z^5+19 a^3 z^3-21 a^3 z+9 a^3 z^{-1} +a^2 z^6-a^2 z^4-7 a^2 z^2-5 a^2 z^{-2} +11 a^2+3 a z^3-8 a z+5 a z^{-1} -2 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 \
-7-6-5-4-3-2-10χ
1       33
-1      242
-3     1 12
-5    12  1
-7   21   1
-9   1    1
-11 12     -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}^{2}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}^{4} {\mathbb Z}^{3}

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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L11n377.gif

L11n377

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L11n379