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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 L11a354's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X8394 X16,11,17,12 X14,8,15,7 X6,16,7,15 X22,17,11,18 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code {1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {4, -1, 2, -5, 6, -4, 7, -9, 8, -10, 11, -7}
A Braid Representative
A Morse Link Presentation L11a354 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+t(2) t(1)-2 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{5/2}+3 q^{3/2}-8 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7+2 z^5 a^5+5 z^3 a^5+3 z a^5-z^7 a^3-3 z^5 a^3-3 z^3 a^3-2 z a^3+2 z^5 a+5 z^3 a+3 z a+a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+2 z^3 a^9-6 z^6 a^8+5 z^4 a^8-2 z^2 a^8-10 z^7 a^7+17 z^5 a^7-16 z^3 a^7+4 z a^7-11 z^8 a^6+21 z^6 a^6-17 z^4 a^6+5 z^2 a^6-7 z^9 a^5+4 z^7 a^5+20 z^5 a^5-23 z^3 a^5+6 z a^5-2 z^{10} a^4-13 z^8 a^4+46 z^6 a^4-39 z^4 a^4+12 z^2 a^4-11 z^9 a^3+24 z^7 a^3-2 z^5 a^3-11 z^3 a^3+4 z a^3-2 z^{10} a^2-5 z^8 a^2+29 z^6 a^2-26 z^4 a^2+6 z^2 a^2-4 z^9 a+9 z^7 a+2 z^5 a-12 z^3 a+6 z a-a z^{-1} -3 z^8+10 z^6-10 z^4+2 z^2+1-z^7 a^{-1} +4 z^5 a^{-1} -6 z^3 a^{-1} +4 z a^{-1} - 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 \
6           11
4          2 -2
2         61 5
0        62  -4
-2       116   5
-4      108    -2
-6     99     0
-8    810      2
-10   59       -4
-12  28        6
-14 15         -4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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