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

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Link Presentations

[edit Notes on L11a333's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,5,21,6 X16,7,17,8 X8,9,1,10 X18,12,19,11 X6,15,7,16 X4,14,5,13 X22,18,9,17 X2,19,3,20 X14,22,15,21
Gauss code {1, -10, 2, -8, 3, -7, 4, -5}, {5, -1, 6, -2, 8, -11, 7, -4, 9, -6, 10, -3, 11, -9}
A Braid Representative
A Morse Link Presentation L11a333 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1) t(2)+1) \left(t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{23}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+4 a^3 z^5+5 a^3 z^3+3 a^3 z+2 a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -13 a z^3+5 z^3 a^{-1} -8 a z+3 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-9 a^2 z^8-7 z^8 a^{-2} -6 z^8-8 a^5 z^7+11 a^3 z^7+37 a z^7+14 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+18 a^4 z^6+32 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +27 z^6-a^7 z^5+13 a^5 z^5-5 a^3 z^5-37 a z^5-9 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-13 a^4 z^4-34 a^2 z^4-9 z^4 a^{-2} +2 z^4 a^{-4} -27 z^4+a^7 z^3-6 a^5 z^3+a^3 z^3+19 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} +4 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +13 z^2+2 a^5 z-4 a^3 z-9 a z-3 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} 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 \
10           1-1
8          3 3
6         61 -5
4        83  5
2       116   -5
0      128    4
-2     1112     1
-4    911      -2
-6   511       6
-8  49        -5
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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).

See/edit the Link_Splice_Base (expert).

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