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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 L11a142's Link Presentations]

Planar diagram presentation X8192 X18,11,19,12 X10,4,11,3 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code {1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, 9, -10, 8, -7, 4, -2, 11, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a142 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-6 u^2 v^3+6 u^2 v^2-2 u^2 v-2 u v^4+8 u v^3-13 u v^2+8 u v-2 u-2 v^3+6 v^2-6 v+2}{u v^2} (db)
Jones polynomial 4 q^{9/2}-\frac{4}{q^{9/2}}-8 q^{7/2}+\frac{8}{q^{7/2}}+13 q^{5/2}-\frac{14}{q^{5/2}}-18 q^{3/2}+\frac{18}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+20 \sqrt{q}-\frac{21}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7+z^7 a^{-1} -a^3 z^5+3 a z^5+3 z^5 a^{-1} -z^5 a^{-3} -2 a^3 z^3+2 a z^3+2 z^3 a^{-1} -2 z^3 a^{-3} -a z+a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -3 z^{10} a^{-2} -3 z^{10}-9 a z^9-15 z^9 a^{-1} -6 z^9 a^{-3} -13 a^2 z^8-2 z^8 a^{-2} -4 z^8 a^{-4} -11 z^8-12 a^3 z^7+11 a z^7+44 z^7 a^{-1} +20 z^7 a^{-3} -z^7 a^{-5} -8 a^4 z^6+23 a^2 z^6+31 z^6 a^{-2} +14 z^6 a^{-4} +48 z^6-4 a^5 z^5+16 a^3 z^5+7 a z^5-35 z^5 a^{-1} -19 z^5 a^{-3} +3 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-14 a^2 z^4-37 z^4 a^{-2} -14 z^4 a^{-4} -44 z^4+2 a^5 z^3-6 a^3 z^3-8 a z^3+8 z^3 a^{-1} +6 z^3 a^{-3} -2 z^3 a^{-5} +4 a^2 z^2+12 z^2 a^{-2} +4 z^2 a^{-4} +12 z^2-a^3 z-a z-a^2+a^3 z^{-1} +a z^{-1} (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 \
12           11
10          3 -3
8         51 4
6        83  -5
4       105   5
2      108    -2
0     1110     1
-2    811      3
-4   610       -4
-6  39        6
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology

(db, data source)

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