L11n396

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L11n395.gif

L11n395

L11n397.gif

L11n397

Contents

L11n396.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n396 at Knotilus!


Link Presentations

[edit Notes on L11n396's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X8493 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X2,14,3,13
Gauss code {1, -11, 5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, -5, -4, 9, -7, 10, 11, -2, 3, 6, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n396 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial q^6-2 q^5+2 q^4-2 q^3+q^2+q+1+3 q^{-1} -2 q^{-2} +2 q^{-3} - q^{-4} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^6 a^{-2} +z^6-a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +6 z^4-3 a^2 z^2-11 z^2 a^{-2} +3 z^2 a^{-4} +11 z^2-2 a^2-6 a^{-2} +2 a^{-4} +6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial a z^9+2 z^9 a^{-1} +z^9 a^{-3} +2 a^2 z^8+5 z^8 a^{-2} +2 z^8 a^{-4} +5 z^8+a^3 z^7-3 a z^7-9 z^7 a^{-1} -3 z^7 a^{-3} +2 z^7 a^{-5} -11 a^2 z^6-33 z^6 a^{-2} -10 z^6 a^{-4} +z^6 a^{-6} -33 z^6-5 a^3 z^5-8 a z^5-4 z^5 a^{-1} -10 z^5 a^{-3} -9 z^5 a^{-5} +17 a^2 z^4+59 z^4 a^{-2} +11 z^4 a^{-4} -4 z^4 a^{-6} +61 z^4+6 a^3 z^3+20 a z^3+32 z^3 a^{-1} +26 z^3 a^{-3} +8 z^3 a^{-5} -14 a^2 z^2-40 z^2 a^{-2} -8 z^2 a^{-4} +2 z^2 a^{-6} -44 z^2-2 a^3 z-10 a z-18 z a^{-1} -14 z a^{-3} -4 z a^{-5} +4 a^2+12 a^{-2} +4 a^{-4} +13-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 \
-5-4-3-2-10123456χ
13           11
11          1 -1
9        121 0
7       121  0
5      232   -1
3     332    2
1    262     2
-1   235      4
-3  12        1
-5 121        0
-7 1          1
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5} {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\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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See/edit the Link Page master template (intermediate).

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L11n395.gif

L11n395

L11n397.gif

L11n397