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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 L11a401's Link Presentations]

Planar diagram presentation X6172 X18,12,19,11 X8493 X2,16,3,15 X16,7,17,8 X22,9,11,10 X4,17,1,18 X10,19,5,20 X12,6,13,5 X14,21,15,22 X20,13,21,14
Gauss code {1, -4, 3, -7}, {9, -1, 5, -3, 6, -8}, {2, -9, 11, -10, 4, -5, 7, -2, 8, -11, 10, -6}
A Braid Representative
A Morse Link Presentation L11a401 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^2 (w-1)^2 \left(w^2-w+1\right)}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-8} +5 q^{-7} -12 q^{-6} +20 q^{-5} -27 q^{-4} +q^3+32 q^{-3} -5 q^2-30 q^{-2} +12 q+28 q^{-1} -19 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^6 z^4-a^6 z^2+2 a^4 z^6+5 a^4 z^4+3 a^4 z^2+a^4 z^{-2} -a^2 z^8-4 a^2 z^6-6 a^2 z^4-3 a^2 z^2-2 a^2 z^{-2} -a^2+z^6+2 z^4+z^2+ z^{-2} +1 (db)
Kauffman polynomial 4 a^4 z^{10}+4 a^2 z^{10}+13 a^5 z^9+24 a^3 z^9+11 a z^9+17 a^6 z^8+28 a^4 z^8+22 a^2 z^8+11 z^8+12 a^7 z^7-6 a^5 z^7-37 a^3 z^7-14 a z^7+5 z^7 a^{-1} +5 a^8 z^6-25 a^6 z^6-73 a^4 z^6-67 a^2 z^6+z^6 a^{-2} -23 z^6+a^9 z^5-14 a^7 z^5-17 a^5 z^5+2 a^3 z^5-4 a z^5-8 z^5 a^{-1} -3 a^8 z^4+14 a^6 z^4+51 a^4 z^4+51 a^2 z^4-z^4 a^{-2} +16 z^4+5 a^7 z^3+10 a^5 z^3+8 a^3 z^3+6 a z^3+3 z^3 a^{-1} -4 a^6 z^2-12 a^4 z^2-12 a^2 z^2-4 z^2-a^3 z-a z+a^4+a^2+1+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 \
7           11
5          4 -4
3         81 7
1        114  -7
-1       178   9
-3      1513    -2
-5     1715     2
-7    1217      5
-9   815       -7
-11  412        8
-13 18         -7
-15 4          4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{17}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{17}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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