L11n358

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L11n357.gif

L11n357

L11n359.gif

L11n359

Contents

L11n358.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n358's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X21,13,22,12 X13,21,14,20 X19,5,20,14 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9}
A Braid Representative
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A Morse Link Presentation L11n358 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(t(2) t(3)^3-t(3)^3+t(2)^2 t(3)^2-2 t(2) t(3)^2+t(1) t(3)-2 t(1) t(2) t(3)-t(1) t(2)^2+t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^3+3 q^2-4 q+7-6 q^{-1} +7 q^{-2} -5 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-2 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^4-2 a^2 z^{-2} -z^2 a^{-2} -2 a^2+z^4+z^2+ z^{-2} +2 (db)
Kauffman polynomial a^3 z^9+a z^9+2 a^4 z^8+5 a^2 z^8+3 z^8+2 a^5 z^7+2 z^7 a^{-1} +a^6 z^6-6 a^4 z^6-22 a^2 z^6-15 z^6-7 a^5 z^5-10 a^3 z^5-11 a z^5-8 z^5 a^{-1} -4 a^6 z^4+4 a^4 z^4+37 a^2 z^4+3 z^4 a^{-2} +32 z^4+5 a^5 z^3+15 a^3 z^3+24 a z^3+15 z^3 a^{-1} +z^3 a^{-3} +4 a^6 z^2-5 a^4 z^2-30 a^2 z^2-5 z^2 a^{-2} -26 z^2-a^5 z-10 a^3 z-14 a z-7 z a^{-1} -2 z a^{-3} -a^6+4 a^4+12 a^2+2 a^{-2} +10+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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 \
-6-5-4-3-2-10123χ
7         1-1
5        2 2
3       32 -1
1      41  3
-1     34   1
-3    43    1
-5   24     2
-7  23      -1
-9 13       2
-11 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=3 {\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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See/edit the Link Page master template (intermediate).

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L11n357

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L11n359