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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 L11a528's Link Presentations]

Planar diagram presentation X8192 X14,3,15,4 X22,18,13,17 X16,11,17,12 X12,15,7,16 X10,22,11,21 X20,10,21,9 X18,6,19,5 X2738 X4,13,5,14 X6,20,1,19
Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 7, -6, 4, -5}, {10, -2, 5, -4, 3, -8, 11, -7, 6, -3}
A Braid Representative
A Morse Link Presentation L11a528 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(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-t(2) t(3) t(1)^2+t(3) t(1)^2-t(1)^2+t(2)^2 t(1)+t(2)^2 t(3)^2 t(1)-2 t(2) t(3)^2 t(1)+t(3)^2 t(1)-2 t(2) t(1)-t(2)^2 t(3) t(1)-t(3) t(1)+t(1)-t(2)^2-t(2)^2 t(3)^2+t(2) t(3)^2+t(2)+t(2)^2 t(3)-t(2) t(3)+t(3)\right)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial -q^6+3 q^5-6 q^4+10 q^3-13 q^2+15 q-14+14 q^{-1} -9 q^{-2} +7 q^{-3} -3 q^{-4} + q^{-5} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +2 z^6-3 a^2 z^4+2 z^4 a^{-2} -z^4 a^{-4} +8 z^4+a^4 z^2-9 a^2 z^2-z^2 a^{-2} -2 z^2 a^{-4} +11 z^2+2 a^4-7 a^2-2 a^{-2} +7+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial 2 a^2 z^{10}+2 z^{10}+3 a^3 z^9+11 a z^9+8 z^9 a^{-1} +a^4 z^8-a^2 z^8+13 z^8 a^{-2} +11 z^8-14 a^3 z^7-44 a z^7-17 z^7 a^{-1} +13 z^7 a^{-3} -5 a^4 z^6-28 a^2 z^6-36 z^6 a^{-2} +10 z^6 a^{-4} -69 z^6+20 a^3 z^5+46 a z^5-8 z^5 a^{-1} -28 z^5 a^{-3} +6 z^5 a^{-5} +10 a^4 z^4+63 a^2 z^4+28 z^4 a^{-2} -14 z^4 a^{-4} +3 z^4 a^{-6} +98 z^4-7 a^3 z^3-5 a z^3+20 z^3 a^{-1} +14 z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} -10 a^4 z^2-46 a^2 z^2-14 z^2 a^{-2} +4 z^2 a^{-4} -54 z^2-3 a^3 z-7 a z-6 z a^{-1} -2 z a^{-3} +5 a^4+13 a^2+4 a^{-2} +13+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 \
13           1-1
11          2 2
9         41 -3
7        62  4
5       74   -3
3      86    2
1     89     1
-1    66      0
-3   510       5
-5  24        -2
-7 15         4
-9 2          -2
-111           1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

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