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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n249 at Knotilus!

Link Presentations

[edit Notes on L11n249's Link Presentations]

Planar diagram presentation X12,1,13,2 X9,19,10,18 X5,14,6,15 X11,6,12,7 X15,11,16,22 X7,20,8,21 X3948 X21,17,22,16 X17,4,18,5 X10,13,1,14 X19,3,20,2
Gauss code {1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}
A Braid Representative
A Morse Link Presentation L11n249 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^3-3 u^3 v^2+2 u^3 v-2 u^2 v^3+7 u^2 v^2-7 u^2 v+u^2+u v^3-7 u v^2+7 u v-2 u+2 v^2-3 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-12 q^{3/2}+\frac{14}{q^{3/2}}+15 \sqrt{q}-\frac{16}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +6 a z^3-5 z^3 a^{-1} +3 a z-3 z a^{-1} (db)
Kauffman polynomial -3 a z^9-3 z^9 a^{-1} -8 a^2 z^8-6 z^8 a^{-2} -14 z^8-8 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 z^7 a^{-3} -3 a^4 z^6+13 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +31 z^6+13 a^3 z^5+27 a z^5+24 z^5 a^{-1} +10 z^5 a^{-3} -3 a^4 z^4-10 a^2 z^4-7 z^4 a^{-2} +2 z^4 a^{-4} -16 z^4-6 a^5 z^3-15 a^3 z^3-18 a z^3-16 z^3 a^{-1} -7 z^3 a^{-3} +a^4 z^2+3 a^2 z^2-z^2 a^{-4} +3 z^2+5 a^5 z+7 a^3 z+4 a z+3 z a^{-1} +z a^{-3} +a^4-a^5 z^{-1} -a^3 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        3 3
6       51 -4
4      73  4
2     85   -3
0    87    1
-2   79     2
-4  57      -2
-6 27       5
-815        -4
-103         3
Integral Khovanov Homology

(db, data source)

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