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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 L11a46's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X20,17,21,18 X18,10,19,9 X8,20,9,19 X4,21,1,22 X14,12,15,11 X10,4,11,3 X12,5,13,6 X22,13,5,14 X2,16,3,15
Gauss code {1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, 5, -3, 6, -10}
A Braid Representative
A Morse Link Presentation L11a46 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{9/2}-\frac{10}{q^{9/2}}-5 q^{7/2}+\frac{16}{q^{7/2}}+10 q^{5/2}-\frac{22}{q^{5/2}}-17 q^{3/2}+\frac{26}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+22 \sqrt{q}-\frac{26}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z+a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -4 a^3 z-2 a^3 z^{-1} - a^{-3} z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-3 z^3 a^{-1} +3 a z+a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-4 a^6 z^4+a^6 z^2+9 a^5 z^7-14 a^5 z^5+11 a^5 z^3-5 a^5 z+a^5 z^{-1} +11 a^4 z^8-15 a^4 z^6+z^6 a^{-4} +9 a^4 z^4-z^4 a^{-4} -3 a^4 z^2+a^4+7 a^3 z^9+7 a^3 z^7+5 z^7 a^{-3} -35 a^3 z^5-10 z^5 a^{-3} +35 a^3 z^3+5 z^3 a^{-3} -15 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^{10}+22 a^2 z^8+9 z^8 a^{-2} -51 a^2 z^6-19 z^6 a^{-2} +36 a^2 z^4+11 z^4 a^{-2} -12 a^2 z^2-2 z^2 a^{-2} +3 a^2+ a^{-2} +14 a z^9+7 z^9 a^{-1} -9 a z^7-2 z^7 a^{-1} -32 a z^5-22 z^5 a^{-1} +36 a z^3+18 z^3 a^{-1} -12 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +2 z^{10}+20 z^8-52 z^6+35 z^4-10 z^2+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 \
10           1-1
8          4 4
6         61 -5
4        114  7
2       116   -5
0      1511    4
-2     1313     0
-4    913      -4
-6   713       6
-8  39        -6
-10 17         6
-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}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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

See/edit the Link_Splice_Base (expert).

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