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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 L11a208's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X20,13,21,14 X10,4,11,3 X16,6,17,5 X4,12,5,11 X22,15,7,16 X12,19,13,20 X14,21,15,22 X2,18,3,17
Gauss code {1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 4, -10, 8, -6, 11, -2, 9, -4, 10, -8}
A Braid Representative
A Morse Link Presentation L11a208 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(v^4-v^3+v^2-v+1\right) (u v-u+1) (u v-v+1)}{u v^3} (db)
Jones polynomial \frac{13}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+3 q^{3/2}-\frac{13}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-6 \sqrt{q}+\frac{8}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^5 z^7-5 a^5 z^5-8 a^5 z^3-5 a^5 z-2 a^5 z^{-1} +a^3 z^9+7 a^3 z^7+18 a^3 z^5+21 a^3 z^3+12 a^3 z+5 a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-6 a z-3 a z^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-6 a^8 z^4+3 a^8 z^2-a^8+6 a^7 z^7-8 a^7 z^5+4 a^7 z^3+6 a^6 z^8-10 a^6 z^6+5 a^6 z^4+2 a^6 z^2+5 a^5 z^9-12 a^5 z^7+15 a^5 z^5-13 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +2 a^4 z^{10}+a^4 z^8-15 a^4 z^6+19 a^4 z^4-12 a^4 z^2+5 a^4+9 a^3 z^9-35 a^3 z^7+50 a^3 z^5-38 a^3 z^3+15 a^3 z-5 a^3 z^{-1} +2 a^2 z^{10}-2 a^2 z^8-12 a^2 z^6+19 a^2 z^4-12 a^2 z^2+5 a^2+4 a z^9-16 a z^7+z^7 a^{-1} +20 a z^5-4 z^5 a^{-1} -14 a z^3+4 z^3 a^{-1} +8 a z-3 a z^{-1} -z a^{-1} +3 z^8-12 z^6+12 z^4-2 z^2 (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 \
6           11
4          2 -2
2         41 3
0        53  -2
-2       83   5
-4      66    0
-6     87     1
-8    56      1
-10   48       -4
-12  25        3
-14 14         -3
-16 2          2
-181           -1
Integral Khovanov Homology

(db, data source)

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