L11n278

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L11n277.gif

L11n277

L11n279.gif

L11n279

Contents

L11n278.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n278 at Knotilus!


Link Presentations

[edit Notes on L11n278's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 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.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n278 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^2-u v^2 w+u v-v w^2+w-1}{\sqrt{u} v w} (db)
Jones polynomial -q^4+q^3-q^2+q+ q^{-1} + q^{-2} + q^{-3} + q^{-4} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^6-a^2 z^4-z^4 a^{-2} +6 z^4-6 a^2 z^2-4 z^2 a^{-2} +11 z^2+2 a^4-9 a^2-3 a^{-2} +10+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2} (db)
Kauffman polynomial z^8 a^{-2} +z^8+a z^7+2 z^7 a^{-1} +z^7 a^{-3} -a^2 z^6-6 z^6 a^{-2} -7 z^6-a^3 z^5-8 a z^5-13 z^5 a^{-1} -6 z^5 a^{-3} +a^4 z^4+7 a^2 z^4+10 z^4 a^{-2} +16 z^4+6 a^3 z^3+21 a z^3+25 z^3 a^{-1} +10 z^3 a^{-3} -5 a^4 z^2-17 a^2 z^2-6 z^2 a^{-2} -18 z^2-11 a^3 z-24 a z-18 z a^{-1} -5 z a^{-3} +7 a^4+15 a^2+3 a^{-2} +12+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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 \
-4-3-2-1012345χ
9         1-1
7          0
5       11 0
3     11   0
1      1   1
-1   131    1
-3  1 1     2
-5  31      2
-71 1       2
-91         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z} {\mathbb Z}^{3} {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\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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L11n279