L11a438

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L11a437.gif

L11a437

L11a439.gif

L11a439

Contents

L11a438.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a438 at Knotilus!


Link Presentations

[edit Notes on L11a438's Link Presentations]

Planar diagram presentation X6172 X2,16,3,15 X10,4,11,3 X14,6,15,5 X22,12,13,11 X12,14,5,13 X4,21,1,22 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9
Gauss code {1, -2, 3, -7}, {4, -1, 9, -10, 11, -3, 5, -6}, {6, -4, 2, -9, 8, -11, 10, -8, 7, -5}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a438 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1)^2 (t(3)-1)^2 (t(2) t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^8+4 q^7-8 q^6+14 q^5-17 q^4+21 q^3-20 q^2+18 q-12+8 q^{-1} -4 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-6 z^2 a^{-2} +6 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2- a^{-4} +1+ a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} (db)
Kauffman polynomial z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +7 z^7 a^{-7} -10 z^5 a^{-7} +2 z^3 a^{-7} +8 z^8 a^{-6} -11 z^6 a^{-6} +3 z^4 a^{-6} +z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +6 z^9 a^{-5} -5 z^7 a^{-5} -z^5 a^{-5} -z^3 a^{-5} +4 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +11 z^8 a^{-4} -35 z^6 a^{-4} +37 z^4 a^{-4} -11 z^2 a^{-4} +2 a^{-4} z^{-2} -4 a^{-4} +12 z^9 a^{-3} -28 z^7 a^{-3} +24 z^5 a^{-3} -7 z^3 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +10 z^8 a^{-2} +a^2 z^6-41 z^6 a^{-2} -2 a^2 z^4+47 z^4 a^{-2} -16 z^2 a^{-2} + a^{-2} z^{-2} - a^{-2} +6 z^9 a^{-1} +4 a z^7-12 z^7 a^{-1} -10 a z^5+4 z^5 a^{-1} +4 a z^3+z^3 a^{-1} +7 z^8-20 z^6+17 z^4-6 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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         51 -4
11        93  6
9       107   -3
7      117    4
5     910     1
3    911      -2
1   511       6
-1  37        -4
-3 15         4
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a437.gif

L11a437

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L11a439