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

Visit L11n418 at Knotilus!

Link Presentations

[edit Notes on L11n418's Link Presentations]

Planar diagram presentation X8192 X7,16,8,17 X3,10,4,11 X2,18,3,17 X18,9,19,10 X11,20,12,21 X5,14,6,15 X15,13,16,22 X13,6,14,1 X19,5,20,4 X21,12,22,7
Gauss code {1, -4, -3, 10, -7, 9}, {-2, -1, 5, 3, -6, 11}, {-9, 7, -8, 2, 4, -5, -10, 6, -11, 8}
A Braid Representative
A Morse Link Presentation L11n418 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v w^3-2 u^2 v w^2+u^2 v w+u^2 w^2-u^2 w+u v^2 w-u v w^3+2 u v w^2-2 u v w+u v-u w^2+v^2 w^2-v^2 w-v w^2+2 v w-v}{u v w^{3/2}} (db)
Jones polynomial -1+3 q^{-1} -4 q^{-2} +7 q^{-3} -6 q^{-4} +7 q^{-5} -5 q^{-6} +4 q^{-7} -2 q^{-8} + q^{-9} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^4-7 a^6 z^2-2 a^6 z^{-2} -8 a^6+a^4 z^6+5 a^4 z^4+10 a^4 z^2+a^4 z^{-2} +7 a^4-a^2 z^4-2 a^2 z^2-a^2 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+4 a^{10} z^2-a^{10}+2 a^9 z^7-7 a^9 z^5+5 a^9 z^3+2 a^8 z^8-6 a^8 z^6+4 a^8 z^4-4 a^8 z^2-a^8 z^{-2} +4 a^8+a^7 z^9-a^7 z^7-4 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +4 a^6 z^8-16 a^6 z^6+28 a^6 z^4-27 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^9-2 a^5 z^7+a^5 z^5+6 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+23 a^4 z^4-23 a^4 z^2-a^4 z^{-2} +10 a^4+a^3 z^7-2 a^3 z^5+6 a^3 z^3-3 a^3 z+3 a^2 z^4-4 a^2 z^2+2 a^2+a z^3-a z (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 \
1         1-1
-1        2 2
-3       32 -1
-5      41  3
-7     34   1
-9    43    1
-11   24     2
-13  23      -1
-15 13       2
-17 1        -1
-191         1
Integral Khovanov Homology

(db, data source)

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