L11n350

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L11n349.gif

L11n349

L11n351.gif

L11n351

Contents

L11n350.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n350 at Knotilus!


Link Presentations

[edit Notes on L11n350's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X11,21,12,20 X7,18,8,19 X9,16,10,17 X17,8,18,9 X19,13,20,22 X13,10,14,11 X21,5,22,12 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, -4, 6, -5, 8, -3, 9}, {-8, -2, 11, 5, -6, 4, -7, 3, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
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A Morse Link Presentation L11n350 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-t(3) t(2)^3+t(2)^3-t(1) t(3)^2 t(2)^2+t(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)+t(1) t(3)^2 t(2)-t(3)^2 t(2)-t(1) t(3) t(2)+t(3) t(2)-t(1) t(3)^3+t(1) t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^5+2 q^4-2 q^3+2 q^2-q+2+ q^{-1} - q^{-2} +2 q^{-3} - q^{-4} + q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^2 a^4+a^4 z^{-2} +2 a^4-z^4 a^2-4 z^2 a^2-2 a^2 z^{-2} -5 a^2+ z^{-2} +2+z^4 a^{-2} +3 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} - a^{-4} (db)
Kauffman polynomial z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +a^4 z^8-7 a^4 z^6+2 z^6 a^{-4} +16 a^4 z^4-7 z^4 a^{-4} -15 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +6 a^4- a^{-4} +a^3 z^9-6 a^3 z^7+z^7 a^{-3} +9 a^3 z^5-3 z^5 a^{-3} -a^3 z^3-5 a^3 z+z a^{-3} +2 a^3 z^{-1} +3 a^2 z^8-21 a^2 z^6+z^6 a^{-2} +44 a^2 z^4-4 z^4 a^{-2} -36 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-7 z^5 a^{-1} +6 a z^3+10 z^3 a^{-1} -8 a z-3 z a^{-1} +2 a z^{-1} +2 z^8-15 z^6+31 z^4-23 z^2- z^{-2} +9 (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 \
-6-5-4-3-2-1012345χ
11           1-1
9          1 1
7         11 0
5       121  0
3      111   1
1     142    1
-1    124     3
-3   11       0
-5  111       1
-7 12         1
-9            0
-111           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2} {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4} {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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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L11n349.gif

L11n349

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L11n351