L11n285

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L11n284.gif

L11n284

L11n286.gif

L11n286

Contents

L11n285.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n285 at Knotilus!


Link Presentations

[edit Notes on L11n285's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X4,15,1,16 X21,18,22,19 X9,21,10,20 X19,5,20,10 X11,16,12,17 X17,22,18,11
Gauss code {1, 4, -3, -6}, {-2, -1, 5, 3, -8, 9}, {-10, 2, -4, -5, 6, 10, -11, 7, -9, 8, -7, 11}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n285 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+2 t(2)+2 t(3)-2}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial 1- q^{-1} +3 q^{-2} -3 q^{-3} +5 q^{-4} -4 q^{-5} +5 q^{-6} -3 q^{-7} +2 q^{-8} - q^{-9} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10}+z^4 a^8+4 z^2 a^8+2 a^8-z^6 a^6-4 z^4 a^6-3 z^2 a^6+a^6 z^{-2} -z^6 a^4-4 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^4 a^2+4 z^2 a^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial a^{11} z^3-2 a^{11} z+2 a^{10} z^4-4 a^{10} z^2+2 a^{10}+a^9 z^7-4 a^9 z^5+8 a^9 z^3-4 a^9 z+2 a^8 z^8-11 a^8 z^6+23 a^8 z^4-16 a^8 z^2+4 a^8+a^7 z^9-4 a^7 z^7+5 a^7 z^5+3 a^6 z^8-14 a^6 z^6+21 a^6 z^4-9 a^6 z^2+a^6 z^{-2} -2 a^6+a^5 z^9-4 a^5 z^7+6 a^5 z^5-8 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +a^4 z^8-2 a^4 z^6-5 a^4 z^4+10 a^4 z^2+2 a^4 z^{-2} -7 a^4+a^3 z^7-3 a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +a^2 z^6-5 a^2 z^4+7 a^2 z^2+a^2 z^{-2} -4 a^2 (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 \
-7-6-5-4-3-2-1012χ
1         11
-1          0
-3       31 2
-5     121  0
-7     42   2
-9   133    1
-11   32     1
-13  13      2
-15 12       -1
-17 1        1
-191         -1
Integral Khovanov Homology

(db, data source)

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