L11n284

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L11n283.gif

L11n283

L11n285.gif

L11n285

Contents

L11n284.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n284's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X10,20,5,19
Gauss code {1, -4, 3, -10}, {2, -1, 5, -3, 6, -11}, {-8, -2, 4, -5, 10, 9, -7, -6, 11, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n284 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(2) t(3)^3+1\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -2 q^7+3 q^6-4 q^5+6 q^4-4 q^3+6 q^2-3 q- q^{-1} +3 (db)
Signature 4 (db)
HOMFLY-PT polynomial z^2 a^{-8} - a^{-8} z^{-2} -z^6 a^{-6} -5 z^4 a^{-6} -5 z^2 a^{-6} +4 a^{-6} z^{-2} +2 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +11 z^4 a^{-4} +6 z^2 a^{-4} -5 a^{-4} z^{-2} -5 a^{-4} -z^6 a^{-2} -4 z^4 a^{-2} -2 z^2 a^{-2} +2 a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -7 z^7 a^{-3} -6 z^7 a^{-5} +2 z^7 a^{-7} -15 z^6 a^{-2} -35 z^6 a^{-4} -20 z^6 a^{-6} -4 z^5 a^{-1} +z^5 a^{-3} -3 z^5 a^{-5} -8 z^5 a^{-7} +21 z^4 a^{-2} +49 z^4 a^{-4} +29 z^4 a^{-6} +z^4 a^{-8} +3 z^3 a^{-1} +4 z^3 a^{-3} +7 z^3 a^{-5} +6 z^3 a^{-7} -9 z^2 a^{-2} -21 z^2 a^{-4} -14 z^2 a^{-6} -2 z^2 a^{-8} +5 z a^{-3} +9 z a^{-5} +5 z a^{-7} +z a^{-9} -3 a^{-2} -4 a^{-4} -2 a^{-6} -5 a^{-3} z^{-1} -9 a^{-5} z^{-1} -5 a^{-7} z^{-1} - a^{-9} z^{-1} +2 a^{-2} z^{-2} +5 a^{-4} z^{-2} +4 a^{-6} z^{-2} + a^{-8} z^{-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 \
-3-2-1012345χ
15        2-2
13       211
11      32 -1
9     321 2
7    35   2
5   321   2
3  25     3
1 11      0
-1 2       2
-31        -1
Integral Khovanov Homology

(db, data source)

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

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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L11n283.gif

L11n283

L11n285.gif

L11n285