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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 L11a516's Link Presentations]

Planar diagram presentation X8192 X16,5,17,6 X18,9,19,10 X10,17,11,18 X20,15,21,16 X14,4,15,3 X4,22,5,21 X2738 X22,12,13,11 X12,14,7,13 X6,19,1,20
Gauss code {1, -8, 6, -7, 2, -11}, {8, -1, 3, -4, 9, -10}, {10, -6, 5, -2, 4, -3, 11, -5, 7, -9}
A Braid Representative
A Morse Link Presentation L11a516 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(3)-1) \left(t(3)^2 t(2)^2+t(1) t(3) t(2)^2-2 t(3) t(2)^2+t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-2 t(1) t(3) t(2)+2 t(3) t(2)-t(2)-t(1) t(3)^2-t(1)+2 t(1) t(3)-t(3)\right)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^4-4 q^3+9 q^2-15 q+21-22 q^{-1} +24 q^{-2} -19 q^{-3} +15 q^{-4} -9 q^{-5} +4 q^{-6} - q^{-7} (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6+3 z^4 a^4+6 z^2 a^4+a^4 z^{-2} +3 a^4-2 z^6 a^2-6 z^4 a^2-7 z^2 a^2-2 a^2 z^{-2} -4 a^2-z^6-z^4+z^2+ z^{-2} +2+z^4 a^{-2} +z^2 a^{-2} (db)
Kauffman polynomial a^7 z^7-3 a^7 z^5+3 a^7 z^3-a^7 z+4 a^6 z^8-13 a^6 z^6+14 a^6 z^4-7 a^6 z^2+2 a^6+6 a^5 z^9-16 a^5 z^7+7 a^5 z^5+6 a^5 z^3-3 a^5 z+3 a^4 z^{10}+7 a^4 z^8-48 a^4 z^6+60 a^4 z^4+z^4 a^{-4} -29 a^4 z^2-a^4 z^{-2} +7 a^4+17 a^3 z^9-44 a^3 z^7+24 a^3 z^5+4 z^5 a^{-3} +3 a^3 z^3-z^3 a^{-3} -4 a^3 z+2 a^3 z^{-1} +3 a^2 z^{10}+19 a^2 z^8-73 a^2 z^6+9 z^6 a^{-2} +74 a^2 z^4-7 z^4 a^{-2} -33 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+11 a z^9-13 a z^7+14 z^7 a^{-1} -8 a z^5-18 z^5 a^{-1} +8 a z^3+7 z^3 a^{-1} -2 a z+2 a z^{-1} +16 z^8-29 z^6+20 z^4-9 z^2- z^{-2} +3 (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 \
9           11
7          3 -3
5         61 5
3        93  -6
1       126   6
-1      1211    -1
-3     1210     2
-5    914      5
-7   610       -4
-9  39        6
-11 16         -5
-13 3          3
-151           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
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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