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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 L11a506's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X14,6,15,5 X10,4,11,3 X4,14,5,13 X2,18,3,17 X18,9,19,10 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code {1, -6, 4, -5, 3, -11}, {2, -1, 7, -4, 9, -8}, {5, -3, 10, -2, 6, -7, 11, -10, 8, -9}
A Braid Representative
A Morse Link Presentation L11a506 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) \left(t(2)^2 t(1)^2+t(2)^2 t(3)^2 t(1)^2-2 t(2) t(3)^2 t(1)^2+t(3)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(3) t(1)^2+2 t(2) t(3) t(1)^2-t(3) t(1)^2-2 t(2)^2 t(1)-t(2)^2 t(3)^2 t(1)+3 t(2) t(3)^2 t(1)-2 t(3)^2 t(1)+3 t(2) t(1)+2 t(2)^2 t(3) t(1)-3 t(2) t(3) t(1)+2 t(3) t(1)-t(1)+t(2)^2-t(2) t(3)^2+t(3)^2-2 t(2)-t(2)^2 t(3)+2 t(2) t(3)-t(3)+1\right)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^7-4 q^6+9 q^5-16 q^4+22 q^3-24 q^2+26 q-21+17 q^{-1} -10 q^{-2} +5 q^{-3} - q^{-4} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-4} +3 z^4 a^{-4} +3 z^2 a^{-4} -z^8 a^{-2} -5 z^6 a^{-2} -a^2 z^4-10 z^4 a^{-2} -a^2 z^2-7 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+ a^{-2} +2 z^6+6 z^4+4 z^2-2 z^{-2} -2 (db)
Kauffman polynomial z^4 a^{-8} +4 z^5 a^{-7} -z^3 a^{-7} +9 z^6 a^{-6} -6 z^4 a^{-6} +z^2 a^{-6} +15 z^7 a^{-5} -20 z^5 a^{-5} +9 z^3 a^{-5} +18 z^8 a^{-4} -34 z^6 a^{-4} +24 z^4 a^{-4} -8 z^2 a^{-4} +13 z^9 a^{-3} +a^3 z^7-17 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +a^3 z^3+10 z^3 a^{-3} +4 z^{10} a^{-2} +5 a^2 z^8+19 z^8 a^{-2} -15 a^2 z^6-77 z^6 a^{-2} +13 a^2 z^4+71 z^4 a^{-2} -2 a^2 z^2-21 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +8 a z^9+21 z^9 a^{-1} -23 a z^7-56 z^7 a^{-1} +16 a z^5+34 z^5 a^{-1} -a z^3-2 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^{10}+6 z^8-49 z^6+53 z^4-14 z^2+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 \
15           11
13          3 -3
11         61 5
9        103  -7
7       126   6
5      1311    -2
3     1311     2
1    1015      5
-1   711       -4
-3  411        7
-5 16         -5
-7 4          4
-91           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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