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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n424 at Knotilus!

Link Presentations

[edit Notes on L11n424's Link Presentations]

Planar diagram presentation X8192 X7,16,8,17 X5,14,6,15 X3,10,4,11 X13,4,14,5 X2,18,3,17 X18,9,19,10 X21,7,22,12 X11,13,12,22 X15,20,16,21 X19,1,20,6
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 L11n424 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) (u w-v-w+1) (u v w-u v-u w+v)}{u v w^{3/2}} (db)
Jones polynomial - q^{-8} +3 q^{-7} -6 q^{-6} +9 q^{-5} -10 q^{-4} +12 q^{-3} -9 q^{-2} +2 q+8 q^{-1} -4 (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^4 a^6-2 z^2 a^6-2 a^6+z^6 a^4+4 z^4 a^4+8 z^2 a^4+a^4 z^{-2} +6 a^4-3 z^4 a^2-8 z^2 a^2-2 a^2 z^{-2} -7 a^2+2 z^2+ z^{-2} +3 (db)
Kauffman polynomial z^5 a^9-2 z^3 a^9+3 z^6 a^8-6 z^4 a^8+z^2 a^8+5 z^7 a^7-12 z^5 a^7+8 z^3 a^7-2 z a^7+5 z^8 a^6-14 z^6 a^6+18 z^4 a^6-12 z^2 a^6+4 a^6+2 z^9 a^5-8 z^5 a^5+14 z^3 a^5-6 z a^5+8 z^8 a^4-28 z^6 a^4+49 z^4 a^4-38 z^2 a^4-a^4 z^{-2} +13 a^4+2 z^9 a^3-4 z^7 a^3+6 z^5 a^3+2 z^3 a^3-7 z a^3+2 a^3 z^{-1} +3 z^8 a^2-11 z^6 a^2+28 z^4 a^2-32 z^2 a^2-2 a^2 z^{-2} +13 a^2+z^7 a+z^5 a-2 z^3 a-3 z a+2 a z^{-1} +3 z^4-7 z^2- z^{-2} +5 (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 \
3         22
1        2 -2
-1       62 4
-3      54  -1
-5     74   3
-7    57    2
-9   45     -1
-11  25      3
-13 14       -3
-15 2        2
-171         -1
Integral Khovanov Homology

(db, data source)

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

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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