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

Visit L11n274 at Knotilus!

Link Presentations

[edit Notes on L11n274's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X4,15,1,16 X19,10,20,5
Gauss code {1, 4, -3, -10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, -4, -5, 10, 9, -7, 6, -11, 8, -9, 7}
A Braid Representative
A Morse Link Presentation L11n274 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2+u v w^3-u v w^2+v w^2-v w-2 w^2+2 w-1}{\sqrt{u} v w^2} (db)
Jones polynomial -q+2-3 q^{-1} +4 q^{-2} -4 q^{-3} +5 q^{-4} -3 q^{-5} +4 q^{-6} - q^{-7} + q^{-8} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^2+2 a^8 z^{-2} +2 a^8-a^6 z^6-6 a^6 z^4-12 a^6 z^2-5 a^6 z^{-2} -12 a^6+a^4 z^8+7 a^4 z^6+18 a^4 z^4+23 a^4 z^2+4 a^4 z^{-2} +14 a^4-a^2 z^6-5 a^2 z^4-7 a^2 z^2-a^2 z^{-2} -4 a^2 (db)
Kauffman polynomial a^{10} z^2-a^{10}+a^9 z^3+3 a^8 z^4-8 a^8 z^2-2 a^8 z^{-2} +8 a^8+2 a^7 z^7-10 a^7 z^5+20 a^7 z^3-18 a^7 z+5 a^7 z^{-1} +3 a^6 z^8-16 a^6 z^6+31 a^6 z^4-33 a^6 z^2-5 a^6 z^{-2} +20 a^6+a^5 z^9-19 a^5 z^5+41 a^5 z^3-33 a^5 z+9 a^5 z^{-1} +5 a^4 z^8-26 a^4 z^6+42 a^4 z^4-32 a^4 z^2-4 a^4 z^{-2} +15 a^4+a^3 z^9-a^3 z^7-14 a^3 z^5+29 a^3 z^3-19 a^3 z+5 a^3 z^{-1} +2 a^2 z^8-10 a^2 z^6+14 a^2 z^4-8 a^2 z^2-a^2 z^{-2} +3 a^2+a z^7-5 a z^5+7 a z^3-4 a z+a z^{-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 \
3         1-1
1        1 1
-1       21 -1
-3     131  1
-5     33   0
-7   142    1
-9  123     2
-11  43      1
-131 2       3
-1522        0
-171         1
Integral Khovanov Homology

(db, data source)

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