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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 L11a467's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X16,8,17,7 X8,16,9,15 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {5, -4, 9, -3, 7, -6, 8, -7}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9}
A Braid Representative
A Morse Link Presentation L11a467 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) \left(-t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-3 t(1) t(2) t(3)^2+5 t(2) t(3)^2-2 t(3)^2-2 t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-2 t(1) t(3)+5 t(1) t(2) t(3)-3 t(2) t(3)+t(3)+t(1) t(2)^2-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial  q^{-6} -q^5-3 q^{-5} +4 q^4+8 q^{-4} -9 q^3-13 q^{-3} +15 q^2+19 q^{-2} -19 q-21 q^{-1} +23 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-3 a^4 z^2+a^4 z^{-2} -z^2 a^{-4} -a^4+3 a^2 z^4+2 z^4 a^{-2} +2 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -2 a^2-z^6-z^4-z^2+ z^{-2} +2 (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-z^3 a^{-5} -a^5 z+5 a^4 z^8-10 a^4 z^6+4 z^6 a^{-4} +7 a^4 z^4-5 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} -a^4 z^{-2} +4 a^4+4 a^3 z^9+a^3 z^7+8 z^7 a^{-3} -18 a^3 z^5-12 z^5 a^{-3} +20 a^3 z^3+7 z^3 a^{-3} -10 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+16 a^2 z^8+9 z^8 a^{-2} -44 a^2 z^6-11 z^6 a^{-2} +46 a^2 z^4+5 z^4 a^{-2} -30 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +12 a^2+2 a^{-2} +9 a z^9+5 z^9 a^{-1} -a z^7+9 z^7 a^{-1} -31 a z^5-33 z^5 a^{-1} +34 a z^3+27 z^3 a^{-1} -14 a z-7 z a^{-1} +2 a z^{-1} +z^{10}+20 z^8-48 z^6+46 z^4-28 z^2- z^{-2} +10 (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 \
11           1-1
9          3 3
7         61 -5
5        93  6
3       117   -4
1      128    4
-1     1012     2
-3    911      -2
-5   511       6
-7  38        -5
-9 16         5
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

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

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