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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 L11a343's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X18,16,19,15 X22,19,11,20 X20,13,21,14 X14,21,15,22 X2,11,3,12 X8,18,9,17
Gauss code {1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 8, -9, 6, -2, 11, -6, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a343 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -3 a^7 z^3-7 a^7 z-5 a^7 z^{-1} +3 a^5 z^5+10 a^5 z^3+14 a^5 z+8 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-8 a^3 z^3-9 a^3 z-4 a^3 z^{-1} +a z^5+2 a z^3+a z (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+5 a^{10} z^2-2 a^{10}+4 a^9 z^7-3 a^9 z^5-2 a^9 z^3+a^9 z+a^9 z^{-1} +4 a^8 z^8+3 a^8 z^6-18 a^8 z^4+22 a^8 z^2-9 a^8+3 a^7 z^9+6 a^7 z^7-17 a^7 z^5+16 a^7 z^3-9 a^7 z+5 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-13 a^6 z^6-12 a^6 z^4+29 a^6 z^2-14 a^6+7 a^5 z^9-27 a^5 z^5+33 a^5 z^3-21 a^5 z+8 a^5 z^{-1} +a^4 z^{10}+12 a^4 z^8-26 a^4 z^6+5 a^4 z^4+13 a^4 z^2-8 a^4+4 a^3 z^9+2 a^3 z^7-24 a^3 z^5+25 a^3 z^3-14 a^3 z+4 a^3 z^{-1} +6 a^2 z^8-12 a^2 z^6+3 a^2 z^4+2 a^2 z^2+4 a z^7-10 a z^5+8 a z^3-2 a z+z^6-2 z^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 \
4           1-1
2          3 3
0         51 -4
-2        93  6
-4       97   -2
-6      117    4
-8     89     1
-10    811      -3
-12   48       4
-14  28        -6
-16 14         3
-18 2          -2
-201           1
Integral Khovanov Homology

(db, data source)

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