L11n367

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L11n366.gif

L11n366

L11n368.gif

L11n368

Contents

L11n367.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n367 at Knotilus!


Link Presentations

[edit Notes on L11n367's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X13,20,14,21 X19,9,20,8 X9,19,10,18 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {-8, 6, -5, 4, -9, 7}, {10, -1, 3, 5, -6, -2, 11, -3, -4, 9, -7, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11n367 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(v^2 w-1\right)}{\sqrt{u} v w} (db)
Jones polynomial -q^4+2 q^3+2 q^{-3} -2 q^2- q^{-2} +3 q+3 q^{-1} -2 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^4 z^{-2} +a^4-a^2 z^4-z^4 a^{-2} -4 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +z^6+5 z^4+7 z^2+ z^{-2} +4 (db)
Kauffman polynomial -a^4 z^2-a^4 z^{-2} +3 a^4+z^7 a^{-3} -5 z^5 a^{-3} +6 z^3 a^{-3} -2 a^3 z-z a^{-3} +2 a^3 z^{-1} +a^2 z^8+2 z^8 a^{-2} -6 a^2 z^6-11 z^6 a^{-2} +12 a^2 z^4+17 z^4 a^{-2} -13 a^2 z^2-9 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -5 a z^7-4 z^7 a^{-1} +6 a z^5+z^5 a^{-1} -a z^3+5 z^3 a^{-1} -4 a z-3 z a^{-1} +2 a z^{-1} +3 z^8-17 z^6+29 z^4-21 z^2- z^{-2} +7 (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 \
-2-1012345χ
9       1-1
7      1 1
5     11 0
3    21  1
1  111   1
-1  32    1
-3112     2
-521      1
-72       2
Integral Khovanov Homology

(db, data source)

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

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L11n366.gif

L11n366

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L11n368