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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 L11a519's Link Presentations]

Planar diagram presentation X8192 X16,6,17,5 X22,20,13,19 X18,10,19,9 X10,21,11,22 X20,11,21,12 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code {1, -7, 8, -9, 2, -11}, {9, -1, 4, -5, 6, -10}, {7, -8, 10, -2, 11, -4, 3, -6, 5, -3}
A Braid Representative
A Morse Link Presentation L11a519 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(2)-1)^2 (t(3)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -q^5-4 q^{-5} +4 q^4+9 q^{-4} -9 q^3-14 q^{-3} +15 q^2+21 q^{-2} -20 q-22 q^{-1} +24 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^6-z^6 a^{-2} -7 a^2 z^4-3 z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +z^8+5 z^6+10 z^4+9 z^2+ z^{-2} +4 (db)
Kauffman polynomial 2 a^2 z^{10}+2 z^{10}+6 a^3 z^9+13 a z^9+7 z^9 a^{-1} +7 a^4 z^8+14 a^2 z^8+10 z^8 a^{-2} +17 z^8+4 a^5 z^7-6 a^3 z^7-20 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-16 a^4 z^6-44 a^2 z^6-16 z^6 a^{-2} +4 z^6 a^{-4} -47 z^6-9 a^5 z^5-7 a^3 z^5+3 a z^5-12 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -2 a^6 z^4+12 a^4 z^4+45 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +48 z^4+5 a^5 z^3+6 a^3 z^3+6 a z^3+12 z^3 a^{-1} +6 z^3 a^{-3} -z^3 a^{-5} +a^6 z^2-6 a^4 z^2-25 a^2 z^2-6 z^2 a^{-2} +z^2 a^{-4} -25 z^2-2 a^3 z-4 a z-3 z a^{-1} -z a^{-3} +3 a^4+7 a^2+2 a^{-2} +7+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 \
11           1-1
9          3 3
7         61 -5
5        93  6
3       116   -5
1      139    4
-1     1113     2
-3    1011      -1
-5   613       7
-7  38        -5
-9 16         5
-11 3          -3
-131           1
Integral Khovanov Homology

(db, data source)

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