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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 L11a517's Link Presentations]

Planar diagram presentation X8192 X14,4,15,3 X12,14,7,13 X2738 X22,10,13,9 X6,22,1,21 X20,16,21,15 X16,5,17,6 X18,11,19,12 X10,17,11,18 X4,19,5,20
Gauss code {1, -4, 2, -11, 8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, 7, -8, 10, -9, 11, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L11a517 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1) \left(t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+2 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+3 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)-1\right)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -q^5-5 q^{-5} +4 q^4+12 q^{-4} -10 q^3-18 q^{-3} +18 q^2+26 q^{-2} -24 q-28 q^{-1} +29 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-6 a^2 z^4-3 z^4 a^{-2} +11 z^4+a^4 z^2-7 a^2 z^2-4 z^2 a^{-2} +10 z^2+a^4-4 a^2- a^{-2} +4+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial 4 a^2 z^{10}+4 z^{10}+11 a^3 z^9+22 a z^9+11 z^9 a^{-1} +11 a^4 z^8+17 a^2 z^8+13 z^8 a^{-2} +19 z^8+5 a^5 z^7-18 a^3 z^7-42 a z^7-10 z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-24 a^4 z^6-58 a^2 z^6-21 z^6 a^{-2} +4 z^6 a^{-4} -58 z^6-8 a^5 z^5+3 a^3 z^5+20 a z^5-4 z^5 a^{-1} -12 z^5 a^{-3} +z^5 a^{-5} -a^6 z^4+15 a^4 z^4+50 a^2 z^4+18 z^4 a^{-2} -4 z^4 a^{-4} +56 z^4+2 a^5 z^3+2 a^3 z^3+8 z^3 a^{-1} +7 z^3 a^{-3} -z^3 a^{-5} -4 a^4 z^2-21 a^2 z^2-9 z^2 a^{-2} +z^2 a^{-4} -27 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         71 -6
5        113  8
3       137   -6
1      1611    5
-1     1415     1
-3    1214      -2
-5   816       8
-7  410        -6
-9 18         7
-11 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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See/edit the Link Page master template (intermediate).

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