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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 L11a416's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X22,14,11,13 X10,15,5,16 X20,17,21,18 X16,7,17,8 X8,20,9,19 X18,10,19,9 X14,22,15,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 6, -7, 8, -4}, {11, -2, 3, -9, 4, -6, 5, -8, 7, -5, 9, -3}
A Braid Representative
A Morse Link Presentation L11a416 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-3 t(1) t(3) t(2)^2+4 t(3) t(2)^2-2 t(2)^2-3 t(1) t(3)^2 t(2)+4 t(3)^2 t(2)-4 t(1) t(2)+8 t(1) t(3) t(2)-8 t(3) t(2)+3 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-4 t(1) t(3)+3 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial - q^{-8} +3 q^{-7} -6 q^{-6} +12 q^{-5} -15 q^{-4} +q^3+19 q^{-3} -4 q^2-18 q^{-2} +8 q+17 q^{-1} -12 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^8+3 a^6 z^2+a^6 z^{-2} +3 a^6-3 a^4 z^4-5 a^4 z^2-2 a^4 z^{-2} -5 a^4+a^2 z^6+2 a^2 z^4+4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} +3 a^2-2 z^4-2 z^2 (db)
Kauffman polynomial a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-6 a^8 z^4+5 a^8 z^2-2 a^8+4 a^7 z^7-3 a^7 z^5-3 a^7 z^3+3 a^7 z+4 a^6 z^8+2 a^6 z^6-15 a^6 z^4+18 a^6 z^2+a^6 z^{-2} -8 a^6+3 a^5 z^9+4 a^5 z^7-9 a^5 z^5+a^5 z^3+5 a^5 z-2 a^5 z^{-1} +a^4 z^{10}+9 a^4 z^8-13 a^4 z^6-4 a^4 z^4+16 a^4 z^2+2 a^4 z^{-2} -9 a^4+7 a^3 z^9-5 a^3 z^7-8 a^3 z^5+4 a^3 z^3+3 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+11 a^2 z^8-27 a^2 z^6+z^6 a^{-2} +15 a^2 z^4-2 z^4 a^{-2} +a^2 z^2+z^2 a^{-2} +a^2 z^{-2} -4 a^2+4 a z^9-a z^7+4 z^7 a^{-1} -13 a z^5-10 z^5 a^{-1} +8 a z^3+6 z^3 a^{-1} +6 z^8-14 z^6+8 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 \
7           11
5          3 -3
3         51 4
1        73  -4
-1       105   5
-3      98    -1
-5     109     1
-7    711      4
-9   58       -3
-11  28        6
-13 14         -3
-15 2          2
-171           -1
Integral Khovanov Homology

(db, data source)

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