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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 L11a42's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X18,8,19,7 X20,13,21,14 X22,15,5,16 X14,21,15,22 X16,19,17,20 X12,10,13,9 X8,18,9,17 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -6, 5, -7, 9, -3, 7, -4, 6, -5}
A Braid Representative
A Morse Link Presentation L11a42 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^4-4 v^3+4 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-\frac{7}{q^{9/2}}-3 q^{7/2}+\frac{12}{q^{7/2}}+6 q^{5/2}-\frac{16}{q^{5/2}}-12 q^{3/2}+\frac{18}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+15 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-6 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +2 a^5 z-7 a^3 z+9 a z-6 z a^{-1} +2 z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-6 a z^9-3 z^9 a^{-1} -5 a^4 z^8-9 a^2 z^8-4 z^8 a^{-2} -8 z^8-5 a^5 z^7-7 a^3 z^7-z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+2 a^4 z^6+14 a^2 z^6+8 z^6 a^{-2} -z^6 a^{-4} +18 z^6-a^7 z^5+7 a^5 z^5+23 a^3 z^5+24 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4+7 a^4 z^4-2 z^4 a^{-2} +3 z^4 a^{-4} -7 z^4+2 a^7 z^3-4 a^5 z^3-24 a^3 z^3-33 a z^3-24 z^3 a^{-1} -9 z^3 a^{-3} -3 a^6 z^2-8 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2-a^7 z+2 a^5 z+13 a^3 z+19 a z+13 z a^{-1} +4 z a^{-3} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 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 \
10           1-1
8          2 2
6         41 -3
4        82  6
2       74   -3
0      118    3
-2     99     0
-4    79      -2
-6   59       4
-8  27        -5
-10 15         4
-12 2          -2
-141           1
Integral Khovanov Homology

(db, data source)

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