L11n390

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L11n389.gif

L11n389

L11n391.gif

L11n391

Contents

L11n390.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n390 at Knotilus!


Link Presentations

[edit Notes on L11n390's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X11,18,12,5 X15,2,16,3
Gauss code {1, 11, -5, -3}, {8, -7, 9, -6}, {-4, -1, 2, 5, 7, -8, -10, 4, 6, -9, -11, -2, 3, 10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n390 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) \left(t(1) t(3)^4+t(1) t(3)^3-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2+t(2) t(3)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} (db)
Jones polynomial  q^{-7} -3 q^{-6} +3 q^{-5} -3 q^{-4} +4 q^{-3} +q^2-2 q^{-2} +2 q^{-1} +1 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^2-a^6 z^{-2} -a^6+4 a^4 z^2+4 a^4 z^{-2} +8 a^4-a^2 z^6-7 a^2 z^4-14 a^2 z^2-5 a^2 z^{-2} -13 a^2+z^4+5 z^2+2 z^{-2} +6 (db)
Kauffman polynomial z^4 a^8-z^2 a^8+3 z^5 a^7-6 z^3 a^7+a^7 z^{-1} +2 z^6 a^6-3 z^4 a^6-a^6 z^{-2} +a^6+3 z^5 a^5-9 z a^5+5 a^5 z^{-1} +6 z^4 a^4-13 z^2 a^4-4 a^4 z^{-2} +12 a^4+z^7 a^3-9 z^5 a^3+28 z^3 a^3-27 z a^3+9 a^3 z^{-1} +z^8 a^2-10 z^6 a^2+31 z^4 a^2-37 z^2 a^2-5 a^2 z^{-2} +21 a^2+z^7 a-9 z^5 a+22 z^3 a-18 z a+5 a z^{-1} +z^8-8 z^6+21 z^4-23 z^2-2 z^{-2} +11 (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 \
-6-5-4-3-2-101234χ
5          11
3          11
1        1  1
-1      3    3
-3     231   0
-5    31     2
-7    21     1
-9  33       0
-11 11        0
-13 2         -2
-151          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{3}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\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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L11n389.gif

L11n389

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L11n391