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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 L11a316's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X6,9,7,10 X16,7,17,8 X8,15,1,16 X20,13,21,14 X18,6,19,5 X4,20,5,19 X22,17,9,18 X14,21,15,22
Gauss code {1, -2, 3, -9, 8, -4, 5, -6}, {4, -1, 2, -3, 7, -11, 6, -5, 10, -8, 9, -7, 11, -10}
A Braid Representative
A Morse Link Presentation L11a316 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+3 u^2 v-u^2-u v^5+3 u v^4-5 u v^3+5 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} (db)
Jones polynomial -\frac{14}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{3}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{16}{q^{13/2}}+\frac{15}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^9 \left(-z^5\right)-3 a^9 z^3-2 a^9 z-a^9 z^{-1} +a^7 z^7+4 a^7 z^5+6 a^7 z^3+6 a^7 z+3 a^7 z^{-1} +a^5 z^7+3 a^5 z^5-4 a^5 z-2 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z (db)
Kauffman polynomial a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+5 a^{12} z^6-5 a^{12} z^4+a^{12} z^2+7 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3-2 a^{11} z+8 a^{10} z^8-18 a^{10} z^6+18 a^{10} z^4-6 a^{10} z^2+a^{10}+6 a^9 z^9-12 a^9 z^7+6 a^9 z^5+a^9 z^3+a^9 z-a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-33 a^8 z^6+39 a^8 z^4-18 a^8 z^2+3 a^8+10 a^7 z^9-33 a^7 z^7+35 a^7 z^5-22 a^7 z^3+11 a^7 z-3 a^7 z^{-1} +2 a^6 z^{10}+a^6 z^8-21 a^6 z^6+26 a^6 z^4-13 a^6 z^2+3 a^6+4 a^5 z^9-13 a^5 z^7+11 a^5 z^5-7 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+11 a^4 z^4-3 a^4 z^2+a^3 z^7-4 a^3 z^5+5 a^3 z^3-2 a^3 z (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 \
0           11
-2          2 -2
-4         51 4
-6        63  -3
-8       84   4
-10      76    -1
-12     98     1
-14    68      2
-16   48       -4
-18  26        4
-20 14         -3
-22 2          2
-241           -1
Integral Khovanov Homology

(db, data source)

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