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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 L11a279's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X14,6,15,5 X4,20,5,19 X18,8,19,7 X6,16,7,15 X22,17,9,18 X16,21,17,22
Gauss code {1, -2, 3, -7, 6, -9, 8, -4}, {4, -1, 2, -3, 5, -6, 9, -11, 10, -8, 7, -5, 11, -10}
A Braid Representative
A Morse Link Presentation L11a279 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^4-t(1)^2 t(2)^3+2 t(1) t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)+1\right)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \frac{12}{q^{9/2}}-\frac{14}{q^{7/2}}-q^{5/2}+\frac{13}{q^{5/2}}+3 q^{3/2}-\frac{12}{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-a^5 z^{-1} +a^3 z^9+7 a^3 z^7+18 a^3 z^5+21 a^3 z^3+11 a^3 z+3 a^3 z^{-1} -a z^7-5 a z^5-8 a z^3-6 a z-2 a z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+6 z^4 a^8-2 z^2 a^8-6 z^7 a^7+9 z^5 a^7-6 z^3 a^7+z a^7-6 z^8 a^6+12 z^6 a^6-12 z^4 a^6+6 z^2 a^6-a^6-5 z^9 a^5+13 z^7 a^5-17 z^5 a^5+14 z^3 a^5-6 z a^5+a^5 z^{-1} -2 z^{10} a^4-z^8 a^4+18 z^6 a^4-27 z^4 a^4+16 z^2 a^4-3 a^4-9 z^9 a^3+36 z^7 a^3-52 z^5 a^3+39 z^3 a^3-15 z a^3+3 a^3 z^{-1} -2 z^{10} a^2+2 z^8 a^2+13 z^6 a^2-20 z^4 a^2+10 z^2 a^2-3 a^2-4 z^9 a+16 z^7 a-19 z^5 a+12 z^3 a-7 z a+2 a z^{-1} -3 z^8+12 z^6-12 z^4+3 z^2-z^7 a^{-1} +4 z^5 a^{-1} -4 z^3 a^{-1} +z a^{-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 \
6           11
4          2 -2
2         41 3
0        42  -2
-2       84   4
-4      76    -1
-6     76     1
-8    57      2
-10   47       -3
-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}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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