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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 L11a290's Link Presentations]

Planar diagram presentation X10,1,11,2 X20,13,21,14 X12,4,13,3 X2,19,3,20 X14,7,15,8 X16,5,17,6 X6,15,7,16 X8,9,1,10 X18,12,19,11 X22,18,9,17 X4,22,5,21
Gauss code {1, -4, 3, -11, 6, -7, 5, -8}, {8, -1, 9, -3, 2, -5, 7, -6, 10, -9, 4, -2, 11, -10}
A Braid Representative
A Morse Link Presentation L11a290 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^4-3 u^3 v^3+3 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+8 u^2 v^3-9 u^2 v^2+4 u^2 v-u^2-u v^5+4 u v^4-9 u v^3+8 u v^2-4 u v+u-v^4+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{23}{q^{5/2}}+\frac{19}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+3 a z^5-z^5 a^{-1} -2 a^5 z^3+2 a^3 z^3+3 a z^3-2 z^3 a^{-1} -2 a^3 z+a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+4 a^7 z^7-10 a^7 z^5+5 a^7 z^3+7 a^6 z^8-18 a^6 z^6+13 a^6 z^4-3 a^6 z^2+7 a^5 z^9-15 a^5 z^7+9 a^5 z^5-a^5 z^3-2 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+6 a^4 z^8-29 a^4 z^6+30 a^4 z^4-8 a^4 z^2-a^4+15 a^3 z^9-37 a^3 z^7+36 a^3 z^5+z^5 a^{-3} -13 a^3 z^3-z^3 a^{-3} -a^3 z+a^3 z^{-1} +3 a^2 z^{10}+9 a^2 z^8-30 a^2 z^6+4 z^6 a^{-2} +29 a^2 z^4-5 z^4 a^{-2} -9 a^2 z^2+z^2 a^{-2} +8 a z^9-10 a z^7+8 z^7 a^{-1} +4 a z^5-12 z^5 a^{-1} -a z^3+5 z^3 a^{-1} -z a^{-1} +10 z^8-16 z^6+9 z^4-3 z^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 \
8           11
6          3 -3
4         61 5
2        93  -6
0       116   5
-2      1210    -2
-4     1110     1
-6    812      4
-8   611       -5
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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