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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 L11a214's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,20,7,19 X16,5,17,6 X18,10,19,9 X4,15,5,16 X10,18,11,17 X14,22,15,21 X20,14,21,13 X2738 X6,11,1,12
Gauss code {1, -10, 2, -6, 4, -11}, {10, -1, 5, -7, 11, -2, 9, -8, 6, -4, 7, -5, 3, -9, 8, -3}
A Braid Representative
A Morse Link Presentation L11a214 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-3 t(1) t(2)^4-4 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+6 t(1)^2 t(2)^2-13 t(1) t(2)^2+6 t(2)^2-4 t(1)^2 t(2)+9 t(1) t(2)-4 t(2)-3 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+9 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+3 a z^3-4 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-4 a^3 z-a z-z a^{-1} +z a^{-3} +a^5 z^{-1} -2 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-6 a^3 z^9-12 a z^9-6 z^9 a^{-1} -7 a^4 z^8-12 a^2 z^8-7 z^8 a^{-2} -12 z^8-6 a^5 z^7+5 a^3 z^7+22 a z^7+7 z^7 a^{-1} -4 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+34 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +40 z^6-a^7 z^5+10 a^5 z^5-a^3 z^5-14 a z^5+7 z^5 a^{-1} +9 z^5 a^{-3} +4 a^6 z^4-2 a^4 z^4-35 a^2 z^4-10 z^4 a^{-2} +2 z^4 a^{-4} -41 z^4+2 a^7 z^3-9 a^5 z^3+9 a z^3-7 z^3 a^{-1} -5 z^3 a^{-3} -a^6 z^2-2 a^4 z^2+16 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} +22 z^2-a^7 z+5 a^5 z-7 a z+z a^{-3} +a^4-3 a^2-2 a^{-2} -5-a^5 z^{-1} +2 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      118    3
-2     1112     1
-4    810      -2
-6   511       6
-8  38        -5
-10  5         5
-1213          -2
-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} {\mathbb Z}
r=-5 {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
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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