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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 L11a147's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X16,7,17,8 X20,13,21,14 X22,15,7,16 X14,21,15,22 X6,18,1,17
Gauss code {1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a147 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^6-2 u^2 v^5+2 u^2 v^4-2 u^2 v^3+2 u^2 v^2-u^2 v-u v^6+3 u v^5-3 u v^4+3 u v^3-3 u v^2+3 u v-u-v^5+2 v^4-2 v^3+2 v^2-2 v+1}{u v^3} (db)
Jones polynomial -\frac{11}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{8}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{5}{q^{17/2}}+\frac{8}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{11}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^5 z^9+a^7 z^7-7 a^5 z^7+a^3 z^7+5 a^7 z^5-17 a^5 z^5+5 a^3 z^5+7 a^7 z^3-16 a^5 z^3+6 a^3 z^3+2 a^7 z-2 a^5 z-a^3 z-a^7 z^{-1} +3 a^5 z^{-1} -2 a^3 z^{-1} (db)
Kauffman polynomial -z^3 a^{13}-3 z^4 a^{12}+z^2 a^{12}-5 z^5 a^{11}+3 z^3 a^{11}-7 z^6 a^{10}+9 z^4 a^{10}-2 z^2 a^{10}-8 z^7 a^9+16 z^5 a^9-7 z^3 a^9+z a^9-7 z^8 a^8+16 z^6 a^8-4 z^4 a^8-3 z^2 a^8+a^8-5 z^9 a^7+13 z^7 a^7-4 z^5 a^7-z^3 a^7-a^7 z^{-1} -2 z^{10} a^6+z^8 a^6+17 z^6 a^6-20 z^4 a^6+z^2 a^6+3 a^6-8 z^9 a^5+38 z^7 a^5-56 z^5 a^5+29 z^3 a^5-z a^5-3 a^5 z^{-1} -2 z^{10} a^4+7 z^8 a^4-z^6 a^4-11 z^4 a^4+3 z^2 a^4+3 a^4-3 z^9 a^3+17 z^7 a^3-31 z^5 a^3+19 z^3 a^3-2 a^3 z^{-1} -z^8 a^2+5 z^6 a^2-7 z^4 a^2+2 z^2 a^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 \
2           1-1
0          2 2
-2         21 -1
-4        62  4
-6       43   -1
-8      75    2
-10     55     0
-12    56      -1
-14   35       2
-16  25        -3
-18 13         2
-20 2          -2
-221           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{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

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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