L11a481

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L11a480.gif

L11a480

L11a482.gif

L11a482

Contents

L11a481.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a481's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X20,16,21,15 X18,10,19,9 X8,18,9,17 X22,20,17,19 X16,22,5,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8}
A Braid Representative
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A Morse Link Presentation L11a481 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(2 u v w^2-4 u v w+2 u v-u w^2+3 u w-4 u-4 v w^2+3 v w-v+2 w^2-4 w+2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -q^8+5 q^7-10 q^6+15 q^5-19 q^4+21 q^3-20 q^2+17 q-10+7 q^{-1} -2 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-z^2 a^{-4} -4 z^2+2 a^2+ a^{-2} - a^{-4} + a^{-6} -3+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial z^5 a^{-9} +5 z^6 a^{-8} -5 z^4 a^{-8} +10 z^7 a^{-7} -15 z^5 a^{-7} +4 z^3 a^{-7} +10 z^8 a^{-6} -11 z^6 a^{-6} -4 z^4 a^{-6} +6 z^2 a^{-6} -2 a^{-6} +5 z^9 a^{-5} +8 z^7 a^{-5} -28 z^5 a^{-5} +18 z^3 a^{-5} -4 z a^{-5} +z^{10} a^{-4} +15 z^8 a^{-4} -24 z^6 a^{-4} -z^4 a^{-4} +12 z^2 a^{-4} -2 a^{-4} +7 z^9 a^{-3} +2 z^7 a^{-3} -30 z^5 a^{-3} +34 z^3 a^{-3} -12 z a^{-3} +z^{10} a^{-2} +8 z^8 a^{-2} +a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4+6 a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-2 a^{-2} +2 z^9 a^{-1} +2 a z^7+6 z^7 a^{-1} -4 a z^5-22 z^5 a^{-1} +20 z^3 a^{-1} +4 a z-4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-4 z^6-2 z^4+6 z^2+2 z^{-2} -5 (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          4 4
13         61 -5
11        94  5
9       117   -4
7      108    2
5     1011     1
3    710      -3
1   512       7
-1  25        -3
-3  5         5
-512          -1
-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} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a480

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L11a482