L10a139

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L10a138.gif

L10a138

L10a140.gif

L10a140

Contents

L10a139.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a139's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X14,8,15,7 X8,14,9,13 X20,15,13,16 X16,19,17,20 X12,18,5,17 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 3, -8}, {5, -4, 6, -7, 8, -3, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
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A Morse Link Presentation L10a139 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(2)-1) \left(t(3) t(2)^2-2 t(2)^2-3 t(1) t(2)+2 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+t(1)-2 t(1) t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} \sqrt{t(3)}} (db)
Jones polynomial  q^{-6} -3 q^{-5} +q^4+6 q^{-4} -2 q^3-8 q^{-3} +5 q^2+10 q^{-2} -7 q-10 q^{-1} +11 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-3 z^2 a^4-2 a^4+2 z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2+z^4-2 z^2-2 z^{-2} -3-2 z^2 a^{-2} + a^{-2} z^{-2} + a^{-4} (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+7 a^5 z^3-2 a^5 z+3 a^4 z^8-5 a^4 z^6-5 a^4 z^4+z^4 a^{-4} +6 a^4 z^2-2 z^2 a^{-4} -a^4+ a^{-4} +a^3 z^9+6 a^3 z^7-24 a^3 z^5+2 z^5 a^{-3} +21 a^3 z^3-2 z^3 a^{-3} -6 a^3 z+6 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} -a^2 z^4-3 z^4 a^{-2} +8 a^2 z^2+3 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-3 a^{-2} +a z^9+6 a z^7+3 z^7 a^{-1} -17 a z^5+11 a z^3-5 z^3 a^{-1} +2 a z+6 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-2 z^6-3 z^4+10 z^2+2 z^{-2} -7 (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 \
-6-5-4-3-2-101234χ
9          11
7         21-1
5        3  3
3       42  -2
1      73   4
-1     56    1
-3    55     0
-5   35      2
-7  35       -2
-9 14        3
-11 2         -2
-131          1
Integral Khovanov Homology

(db, data source)

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