L11a238

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L11a237.gif

L11a237

L11a239.gif

L11a239

Contents

L11a238.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a238 at Knotilus!


Link Presentations

[edit Notes on L11a238's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X22,12,7,11 X16,9,17,10 X14,22,15,21 X10,15,11,16 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code {1, -9, 2, -10, 7, -11}, {9, -1, 4, -6, 3, -2, 10, -5, 6, -4, 8, -7, 11, -8, 5, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a238 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+7 t(1)^2 t(2)^2-11 t(1) t(2)^2+7 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{17/2}-4 q^{15/2}+9 q^{13/2}-15 q^{11/2}+19 q^{9/2}-22 q^{7/2}+21 q^{5/2}-18 q^{3/2}+13 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +z a^{-7} -2 z^5 a^{-5} -4 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +4 z^3 a^{-3} +4 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-5 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -7 z^4 a^{-8} +2 z^2 a^{-8} +14 z^7 a^{-7} -20 z^5 a^{-7} +12 z^3 a^{-7} -3 z a^{-7} +14 z^8 a^{-6} -21 z^6 a^{-6} +8 z^4 a^{-6} -z^2 a^{-6} +8 z^9 a^{-5} +z^7 a^{-5} -33 z^5 a^{-5} +29 z^3 a^{-5} -9 z a^{-5} + a^{-5} z^{-1} +2 z^{10} a^{-4} +18 z^8 a^{-4} -55 z^6 a^{-4} +39 z^4 a^{-4} -8 z^2 a^{-4} +12 z^9 a^{-3} -22 z^7 a^{-3} -10 z^5 a^{-3} +28 z^3 a^{-3} -13 z a^{-3} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +7 z^8 a^{-2} -35 z^6 a^{-2} +34 z^4 a^{-2} -9 z^2 a^{-2} + a^{-2} +4 z^9 a^{-1} +a z^7-8 z^7 a^{-1} -4 a z^5-5 z^5 a^{-1} +6 a z^3+18 z^3 a^{-1} -4 a z-11 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +3 z^8-10 z^6+11 z^4-4 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 \
-4-3-2-101234567χ
18           1-1
16          3 3
14         61 -5
12        93  6
10       106   -4
8      129    3
6     1011     1
4    811      -3
2   611       5
0  27        -5
-2 16         5
-4 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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L11a239