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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 L11a456's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X20,11,21,12 X18,8,19,7 X16,10,17,9 X8,18,9,17 X22,19,13,20 X10,14,11,13 X12,21,5,22 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 4, -6, 5, -8, 3, -9}, {8, -2, 11, -5, 6, -4, 7, -3, 9, -7}
A Braid Representative
A Morse Link Presentation L11a456 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^3 w^2-u v^3 w+u v^2 w^3-4 u v^2 w^2+5 u v^2 w-u v^2-2 u v w^3+5 u v w^2-4 u v w+u v+u w^3-2 u w^2+u w-v^3 w^2+2 v^3 w-v^3-v^2 w^3+4 v^2 w^2-5 v^2 w+2 v^2+v w^3-5 v w^2+4 v w-v+w^2-w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial q^6-3 q^5- q^{-5} +7 q^4+3 q^{-4} -11 q^3-7 q^{-3} +17 q^2+13 q^{-2} -18 q-16 q^{-1} +19 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} -a^4 z^2+2 z^2 a^{-4} + a^{-4} z^{-2} -a^4+2 a^{-4} -z^6 a^{-2} +2 a^2 z^4-3 z^4 a^{-2} +3 a^2 z^2-6 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-5 a^{-2} -z^6-z^4+z^2+ z^{-2} +2 (db)
Kauffman polynomial z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} -2 a^5 z^3+5 z^3 a^{-5} +a^5 z+5 z^8 a^{-4} +3 a^4 z^6-14 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +3 a^4 z^2-13 z^2 a^{-4} - a^{-4} z^{-2} -a^4+6 a^{-4} +4 z^9 a^{-3} +5 a^3 z^7-5 z^7 a^{-3} -6 a^3 z^5-3 z^5 a^{-3} +a^3 z^3+6 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +6 a^2 z^8+12 z^8 a^{-2} -7 a^2 z^6-41 z^6 a^{-2} +3 a^2 z^4+55 z^4 a^{-2} -a^2 z^2-39 z^2 a^{-2} -2 a^{-2} z^{-2} +13 a^{-2} +4 a z^9+8 z^9 a^{-1} +2 a z^7-11 z^7 a^{-1} -12 a z^5+10 a z^3+8 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+13 z^8-36 z^6+44 z^4-28 z^2- z^{-2} +9 (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 \
13           11
11          2 -2
9         51 4
7        73  -4
5       104   6
3      87    -1
1     1110     1
-1    811      3
-3   58       -3
-5  28        6
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

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