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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 L11n394's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X9,22,10,19 X3849 X21,17,22,16 X11,5,12,18 X5,21,6,20 X17,11,18,10 X19,12,20,13 X13,2,14,3
Gauss code {1, 11, -5, -3}, {-10, 8, -6, 4}, {-8, -1, 2, 5, -4, 9, -7, 10, -11, -2, 3, 6, -9, 7}
A Braid Representative
A Morse Link Presentation L11n394 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) (w-1) \left(w^2-w+1\right) \left(u w^2-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial - q^{-6} +4 q^{-5} -4 q^{-4} +q^3+8 q^{-3} -3 q^2-8 q^{-2} +5 q+8 q^{-1} -6 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^2 z^8+a^4 z^6-6 a^2 z^6+z^6+5 a^4 z^4-12 a^2 z^4+4 z^4-a^6 z^2+7 a^4 z^2-10 a^2 z^2+4 z^2-a^6+2 a^4-3 a^2+2+a^6 z^{-2} -2 a^4 z^{-2} +a^2 z^{-2} (db)
Kauffman polynomial a^7 z^3-a^7 z+4 a^6 z^4-6 a^6 z^2+a^6 z^{-2} -a^6+2 a^5 z^7-6 a^5 z^5+9 a^5 z^3-a^5 z-2 a^5 z^{-1} +4 a^4 z^8-19 a^4 z^6+37 a^4 z^4-25 a^4 z^2+2 a^4 z^{-2} +2 a^4+2 a^3 z^9-4 a^3 z^7-6 a^3 z^5+19 a^3 z^3-5 a^3 z-2 a^3 z^{-1} +8 a^2 z^8-35 a^2 z^6+z^6 a^{-2} +51 a^2 z^4-3 z^4 a^{-2} -28 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} +5 a^2+2 a z^9-3 a z^7+3 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +17 a z^3+6 z^3 a^{-1} -7 a z-2 z a^{-1} +4 z^8-15 z^6+15 z^4-8 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 \
7         11
5        2 -2
3       31 2
1      32  -1
-1    163   2
-3    55    0
-5   44     0
-7 125      4
-9 33       0
-11 3        3
-131         -1
Integral Khovanov Homology

(db, data source)

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