L11n345

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L11n344.gif

L11n344

L11n346.gif

L11n346

Contents

L11n345.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n345's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X11,19,12,18 X16,8,17,7 X17,21,18,20 X19,5,20,12 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, {9, -2, 11, -4, -5, 3, -6, 5, 7, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n345 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3+t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3) t(2)^2-t(3)^2 t(2)+t(2)+t(1)-t(1) t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^{10}-q^8+q^7-q^6+3 q^5-2 q^4+3 q^3-q^2+q (db)
Signature 6 (db)
HOMFLY-PT polynomial -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -20 z^2 a^{-6} +9 z^2 a^{-8} -z^2 a^{-10} +7 a^{-4} -11 a^{-6} +4 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} (db)
Kauffman polynomial z^4 a^{-12} -4 z^2 a^{-12} + a^{-12} -z^3 a^{-11} -z^4 a^{-10} -z^2 a^{-10} +2 z^7 a^{-9} -11 z^5 a^{-9} +11 z^3 a^{-9} +3 z^8 a^{-8} -18 z^6 a^{-8} +29 z^4 a^{-8} -16 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} -3 z^7 a^{-7} -7 z^5 a^{-7} +20 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -25 z^6 a^{-6} +48 z^4 a^{-6} -37 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +8 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} (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 \
-2-1012345678χ
21          11
19          11
17       21  -1
15      11   0
13     231   0
11    112    2
9   12      1
7  21       1
5 13        2
3           0
11          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5 i=7
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=6 {\mathbb Z}_2 {\mathbb Z}
r=7
r=8 {\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

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See/edit the Link Page master template (intermediate).

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L11n344

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L11n346