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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 L10a124's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,11,19,12 X20,17,9,18 X16,14,17,13 X8,16,5,15 X14,8,15,7 X12,19,13,20 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 7, -6}, {10, -2, 3, -8, 5, -7, 6, -5, 4, -3, 8, -4}
A Braid Representative
A Morse Link Presentation L10a124 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^3-3 u v w^2+3 u v w-u v-u w^3+4 u w^2-4 u w+2 u-2 v w^3+4 v w^2-4 v w+v+w^3-3 w^2+3 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -q^2+4 q-7+11 q^{-1} -11 q^{-2} +14 q^{-3} -11 q^{-4} +9 q^{-5} -5 q^{-6} +2 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^8 z^{-2} -a^8+3 a^6 z^2+4 a^6 z^{-2} +6 a^6-3 a^4 z^4-8 a^4 z^2-5 a^4 z^{-2} -10 a^4+a^2 z^6+3 a^2 z^4+5 a^2 z^2+2 a^2 z^{-2} +5 a^2-z^4-z^2 (db)
Kauffman polynomial z^5 a^9-3 z^3 a^9+3 z a^9-a^9 z^{-1} +2 z^6 a^8-4 z^4 a^8+3 z^2 a^8+a^8 z^{-2} -2 a^8+2 z^7 a^7+2 z^5 a^7-12 z^3 a^7+13 z a^7-5 a^7 z^{-1} +2 z^8 a^6+3 z^6 a^6-11 z^4 a^6+14 z^2 a^6+4 a^6 z^{-2} -10 a^6+z^9 a^5+5 z^7 a^5-4 z^5 a^5-12 z^3 a^5+21 z a^5-9 a^5 z^{-1} +6 z^8 a^4-3 z^6 a^4-12 z^4 a^4+19 z^2 a^4+5 a^4 z^{-2} -14 a^4+z^9 a^3+9 z^7 a^3-16 z^5 a^3+11 z a^3-5 a^3 z^{-1} +4 z^8 a^2-12 z^4 a^2+11 z^2 a^2+2 a^2 z^{-2} -7 a^2+6 z^7 a-10 z^5 a+2 z^3 a+4 z^6-7 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} (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 \
5          1-1
3         3 3
1        41 -3
-1       73  4
-3      77   0
-5     74    3
-7    47     3
-9   57      -2
-11  15       4
-13 14        -3
-15 1         1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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