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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 L11a220's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X20,15,21,16 X18,21,19,22 X14,19,15,20 X2738 X4,11,5,12 X6,18,1,17
Gauss code {1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, -5, 11, -7, 8, -6, 7, -3}
A Braid Representative
A Morse Link Presentation L11a220 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-4 t(1)^2 t(2)^3+8 t(1) t(2)^3-5 t(2)^3+7 t(1)^2 t(2)^2-13 t(1) t(2)^2+7 t(2)^2-5 t(1)^2 t(2)+8 t(1) t(2)-4 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -\frac{14}{q^{9/2}}-q^{7/2}+\frac{18}{q^{7/2}}+4 q^{5/2}-\frac{23}{q^{5/2}}-9 q^{3/2}+\frac{22}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{8}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+5 z a^5+3 a^5 z^{-1} -3 z^5 a^3-8 z^3 a^3-8 z a^3-2 a^3 z^{-1} +z^7 a+4 z^5 a+8 z^3 a+5 z a-z^5 a^{-1} -2 z^3 a^{-1} -2 z a^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-9 a^3 z^9-5 a z^9-5 a^6 z^8-16 a^4 z^8-20 a^2 z^8-9 z^8-3 a^7 z^7-2 a^5 z^7-a^3 z^7-10 a z^7-8 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+39 a^4 z^6+43 a^2 z^6-4 z^6 a^{-2} +10 z^6+7 a^7 z^5+20 a^5 z^5+35 a^3 z^5+35 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-3 a^6 z^4-27 a^4 z^4-28 a^2 z^4+5 z^4 a^{-2} -2 z^4-6 a^7 z^3-20 a^5 z^3-33 a^3 z^3-28 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-3 a^6 z^2+4 a^4 z^2+6 a^2 z^2-2 z^2 a^{-2} +3 a^7 z+9 a^5 z+10 a^3 z+7 a z+3 z a^{-1} +a^8+3 a^6+3 a^4-a^7 z^{-1} -3 a^5 z^{-1} -2 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 \
8           11
6          3 -3
4         61 5
2        93  -6
0       116   5
-2      1210    -2
-4     1110     1
-6    813      5
-8   610       -4
-10  28        6
-12 16         -5
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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