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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 L11a356's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,5,17,6 X14,3,15,4 X20,10,21,9 X18,8,19,7 X22,17,11,18 X8,20,9,19 X4,15,5,16 X6,22,7,21 X2,11,3,12 X10,13,1,14
Gauss code {1, -10, 3, -8, 2, -9, 5, -7, 4, -11}, {10, -1, 11, -3, 8, -2, 6, -5, 7, -4, 9, -6}
A Braid Representative
A Morse Link Presentation L11a356 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^4 v^3-u^4 v^2+u^3 v^4-3 u^3 v^3+4 u^3 v^2-2 u^3 v-u^2 v^4+4 u^2 v^3-7 u^2 v^2+4 u^2 v-u^2-2 u v^3+4 u v^2-3 u v+u-v^2+v}{u^2 v^2} (db)
Jones polynomial -\frac{8}{q^{9/2}}-q^{7/2}+\frac{10}{q^{7/2}}+3 q^{5/2}-\frac{13}{q^{5/2}}-6 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+9 \sqrt{q}-\frac{12}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+5 a^3 z^5+4 a z^5-z^5 a^{-1} -4 a^5 z^3+9 a^3 z^3+4 a z^3-3 z^3 a^{-1} -4 a^5 z+6 a^3 z-a z-2 z a^{-1} +a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial a^8 z^6-4 a^8 z^4+4 a^8 z^2+2 a^7 z^7-7 a^7 z^5+7 a^7 z^3-2 a^7 z+2 a^6 z^8-4 a^6 z^6+a^6 z^2+2 a^5 z^9-5 a^5 z^7+9 a^5 z^5-12 a^5 z^3+3 a^5 z+a^4 z^{10}-a^4 z^6+2 a^4 z^4-4 a^4 z^2+5 a^3 z^9-16 a^3 z^7+30 a^3 z^5+z^5 a^{-3} -27 a^3 z^3-2 z^3 a^{-3} +10 a^3 z-a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +16 a^2 z^4-6 z^4 a^{-2} -6 a^2 z^2+z^2 a^{-2} +a^2+3 a z^9-4 a z^7+5 z^7 a^{-1} +a z^5-12 z^5 a^{-1} +3 a z^3+9 z^3 a^{-1} +2 a z-3 z a^{-1} -a z^{-1} +5 z^8-12 z^6+12 z^4-4 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          2 -2
4         41 3
2        52  -3
0       74   3
-2      76    -1
-4     66     0
-6    58      3
-8   35       -2
-10  15        4
-12 13         -2
-14 1          1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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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