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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 L11a237's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X14,12,15,11 X22,15,7,16 X16,9,17,10 X10,21,11,22 X18,6,19,5 X20,18,21,17 X2738 X4,14,5,13 X6,20,1,19
Gauss code {1, -9, 2, -10, 7, -11}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -5, 8, -7, 11, -8, 6, -4}
A Braid Representative
A Morse Link Presentation L11a237 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u+v^4-5 v^3+8 v^2-5 v+1}{u v^2} (db)
Jones polynomial -11 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{3}{q^{7/2}}-23 q^{5/2}+\frac{8}{q^{5/2}}+25 q^{3/2}-\frac{15}{q^{3/2}}-q^{13/2}+5 q^{11/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} -z^3 a^{-5} -2 a^3 z+6 a z-4 z a^{-1} +z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-5 a z^9-13 z^9 a^{-1} -8 z^9 a^{-3} -5 a^2 z^8-21 z^8 a^{-2} -13 z^8 a^{-4} -13 z^8-3 a^3 z^7+2 a z^7+12 z^7 a^{-1} -4 z^7 a^{-3} -11 z^7 a^{-5} -a^4 z^6+8 a^2 z^6+46 z^6 a^{-2} +17 z^6 a^{-4} -5 z^6 a^{-6} +33 z^6+7 a^3 z^5+12 a z^5+16 z^5 a^{-1} +27 z^5 a^{-3} +15 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-a^2 z^4-25 z^4 a^{-2} -4 z^4 a^{-4} +4 z^4 a^{-6} -21 z^4-6 a^3 z^3-16 a z^3-21 z^3 a^{-1} -16 z^3 a^{-3} -5 z^3 a^{-5} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} +3 a^3 z+10 a z+9 z a^{-1} +2 z a^{-3} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} 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 \
14           11
12          4 -4
10         71 6
8        104  -6
6       137   6
4      1311    -2
2     1212     0
0    914      5
-2   611       -5
-4  29        7
-6 16         -5
-8 2          2
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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