L11n241

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L11n240.gif

L11n240

L11n242.gif

L11n242

Contents

L11n241.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n241 at Knotilus!


Link Presentations

[edit Notes on L11n241's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,8,17,7 X5,1,6,10 X3746 X9,5,10,4 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X2,11,3,12 X8,18,9,17
Gauss code {1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, 6, -9, 8, -2, 11, -6, 7, -8, 9, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n241 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-t(2) t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial 7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{2}{q^{3/2}}+2 q^{13/2}-5 q^{11/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z a^{-7} + a^{-7} z^{-1} -z^5 a^{-5} -4 z^3 a^{-5} -6 z a^{-5} -4 a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +12 z a^{-3} +6 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-10 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} (db)
Kauffman polynomial 3 z^2 a^{-8} - a^{-8} +z^5 a^{-7} +5 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +5 z^6 a^{-6} -9 z^4 a^{-6} +9 z^2 a^{-6} -3 a^{-6} +8 z^7 a^{-5} -25 z^5 a^{-5} +31 z^3 a^{-5} -18 z a^{-5} +4 a^{-5} z^{-1} +5 z^8 a^{-4} -10 z^6 a^{-4} -2 z^4 a^{-4} +8 z^2 a^{-4} -3 a^{-4} +z^9 a^{-3} +10 z^7 a^{-3} -47 z^5 a^{-3} +58 z^3 a^{-3} -31 z a^{-3} +6 a^{-3} z^{-1} +7 z^8 a^{-2} -23 z^6 a^{-2} +16 z^4 a^{-2} - a^{-2} +z^9 a^{-1} +a z^7+3 z^7 a^{-1} -5 a z^5-26 z^5 a^{-1} +9 a z^3+41 z^3 a^{-1} -7 a z-24 z a^{-1} +2 a z^{-1} +5 a^{-1} z^{-1} +2 z^8-8 z^6+9 z^4-2 z^2-1 (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
14         2-2
12        3 3
10       42 -2
8      53  2
6     44   0
4    55    0
2   46     2
0  13      -2
-2 14       3
-4 1        -1
-61         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}_2^{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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L11n240.gif

L11n240

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L11n242