L10a126

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L10a125.gif

L10a125

L10a127.gif

L10a127

Contents

L10a126.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a126's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,12,9,11 X18,14,19,13 X14,18,15,17 X12,20,13,19 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 3, -4}, {10, -2, 5, -8, 6, -7, 4, -3, 7, -6, 8, -5}
A Braid Representative
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A Morse Link Presentation L10a126 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-3 t(2) t(1)+2 t(2) t(3) t(1)-3 t(3) t(1)+3 t(1)+3 t(2)-3 t(2) t(3)+3 t(3)-2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} (db)
Jones polynomial  q^{-7} - q^{-6} +4 q^{-5} -4 q^{-4} +q^3+7 q^{-3} -3 q^2-7 q^{-2} +4 q+7 q^{-1} -5 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+3 z^2 a^4+a^4 z^{-2} +3 a^4-z^4 a^2-z^4-z^2+z^2 a^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8+a^5 z^7-2 a^3 z^7+3 z^7 a^{-1} +a^6 z^6-13 a^2 z^6+z^6 a^{-2} -11 z^6+a^7 z^5+a^5 z^5+4 a^3 z^5-7 a z^5-11 z^5 a^{-1} +a^8 z^4+2 a^6 z^4+a^4 z^4+13 a^2 z^4-3 z^4 a^{-2} +10 z^4-3 a^3 z^3+5 a z^3+8 z^3 a^{-1} -3 a^8 z^2-6 a^6 z^2-3 a^4 z^2-3 a^2 z^2+z^2 a^{-2} -2 z^2-3 a^7 z-3 a^5 z+3 a^8+5 a^6+3 a^4+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^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 \
-6-5-4-3-2-101234χ
7          11
5         2 -2
3        21 1
1       32  -1
-1      42   2
-3     44    0
-5    33     0
-7   14      3
-9  33       0
-11 14        3
-13           0
-151          1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

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