L11n269

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L11n268.gif

L11n268

L11n270.gif

L11n270

Contents

L11n269.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n269's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X20,13,21,14 X12,19,13,20 X14,21,15,22 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n269 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(3)^5-t(1) t(3)^4-t(2) t(3)^4+t(1) t(3)^3+t(2) t(3)^3-t(1) t(3)^2-t(2) t(3)^2+t(1) t(3)+t(2) t(3)-t(1) t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} (db)
Jones polynomial - q^{-8} + q^{-7} -2 q^{-6} +2 q^{-5} -3 q^{-4} +3 q^{-3} +q^2+2 q^{-1} +1 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 \left(-z^4\right)-4 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^6+7 a^4 z^4+18 a^4 z^2+7 a^4 z^{-2} +18 a^4-a^2 z^6-8 a^2 z^4-20 a^2 z^2-8 a^2 z^{-2} -20 a^2+z^4+5 z^2+3 z^{-2} +7 (db)
Kauffman polynomial a^9 z^5-4 a^9 z^3+3 a^9 z-a^9 z^{-1} +a^8 z^6-3 a^8 z^4+a^8+a^7 z^7-3 a^7 z^5+3 a^7 z-a^7 z^{-1} +a^6 z^8-5 a^6 z^6+9 a^6 z^4-9 a^6 z^2-2 a^6 z^{-2} +7 a^6+2 a^5 z^7-12 a^5 z^5+26 a^5 z^3-21 a^5 z+7 a^5 z^{-1} +a^4 z^8-8 a^4 z^6+24 a^4 z^4-32 a^4 z^2-7 a^4 z^{-2} +22 a^4+2 a^3 z^7-17 a^3 z^5+46 a^3 z^3-45 a^3 z+15 a^3 z^{-1} +a^2 z^8-10 a^2 z^6+33 a^2 z^4-46 a^2 z^2-8 a^2 z^{-2} +28 a^2+a z^7-9 a z^5+24 a z^3-24 a z+8 a z^{-1} +z^8-8 z^6+21 z^4-23 z^2-3 z^{-2} +13 (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 \
-7-6-5-4-3-2-101234χ
5           11
3           11
1         1  1
-1       3    3
-3      141   2
-5     3      3
-7    121     0
-9   23       -1
-11   11       0
-13 12         -1
-15            0
-171           -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}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}^{3}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\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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L11n268

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L11n270