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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 L11a441's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X12,15,5,16 X22,17,13,18 X16,7,17,8 X8,22,9,21 X20,12,21,11 X18,10,19,9 X10,20,11,19 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 5, -6, 8, -9, 7, -3}, {11, -2, 3, -5, 4, -8, 9, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L11a441 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-t(3) t(2)^3-t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-6 t(3)^2 t(2)^2+t(1) t(2)^2-4 t(1) t(3) t(2)^2+4 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-t(3)^3 t(2)-4 t(1) t(3)^2 t(2)+4 t(3)^2 t(2)-2 t(1) t(2)+6 t(1) t(3) t(2)-4 t(3) t(2)+t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial - q^{-7} +3 q^{-6} -6 q^{-5} +q^4+11 q^{-4} -4 q^3-14 q^{-3} +9 q^2+19 q^{-2} -13 q-18 q^{-1} +17 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 \left(-z^2\right)-2 a^6+3 a^4 z^4+9 a^4 z^2+a^4 z^{-2} +7 a^4-2 a^2 z^6-8 a^2 z^4+z^4 a^{-2} -13 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -9 a^2-z^6-z^4+3 z^2+ z^{-2} +4 (db)
Kauffman polynomial a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+8 a^3 z^9+5 a z^9+3 a^6 z^8+9 a^4 z^8+16 a^2 z^8+10 z^8+a^7 z^7-6 a^5 z^7-13 a^3 z^7+6 a z^7+12 z^7 a^{-1} -12 a^6 z^6-41 a^4 z^6-48 a^2 z^6+9 z^6 a^{-2} -10 z^6-4 a^7 z^5-6 a^5 z^5-11 a^3 z^5-29 a z^5-16 z^5 a^{-1} +4 z^5 a^{-3} +16 a^6 z^4+55 a^4 z^4+47 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -2 z^4+5 a^7 z^3+17 a^5 z^3+24 a^3 z^3+20 a z^3+7 z^3 a^{-1} -z^3 a^{-3} -9 a^6 z^2-34 a^4 z^2-29 a^2 z^2+4 z^2 a^{-2} -2 a^7 z-7 a^5 z-12 a^3 z-8 a z-z a^{-1} +3 a^6+11 a^4+11 a^2- a^{-2} +3+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 \
9           11
7          3 -3
5         61 5
3        73  -4
1       106   4
-1      1110    -1
-3     87     1
-5    611      5
-7   58       -3
-9  27        5
-11 14         -3
-13 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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

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