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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 L10a125's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X20,14,9,13 X18,12,19,11 X12,20,13,19 X14,18,15,17 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 3, -4}, {10, -2, 6, -7, 5, -8, 4, -3, 8, -6, 7, -5}
A Braid Representative
A Morse Link Presentation L10a125 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w^3-2 u v w^2+2 u v w-2 u v-2 u w^3+3 u w^2-3 u w+2 u-2 v w^3+3 v w^2-3 v w+2 v+2 w^3-2 w^2+2 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial q^4-4 q^3+7 q^2-10 q+11-11 q^{-1} +11 q^{-2} -6 q^{-3} +5 q^{-4} - q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial 2 a^6 z^{-2} +a^6-3 z^2 a^4-5 a^4 z^{-2} -8 a^4+3 z^4 a^2+9 z^2 a^2+4 a^2 z^{-2} +10 a^2-z^6-3 z^4-4 z^2- z^{-2} -3+z^4 a^{-2} +z^2 a^{-2} (db)
Kauffman polynomial a^6 z^6-5 a^6 z^4+9 a^6 z^2+2 a^6 z^{-2} -7 a^6+a^5 z^7-a^5 z^5-6 a^5 z^3+11 a^5 z-5 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-14 a^4 z^4+z^4 a^{-4} +20 a^4 z^2+5 a^4 z^{-2} -14 a^4+a^3 z^9+3 a^3 z^5+4 z^5 a^{-3} -17 a^3 z^3-3 z^3 a^{-3} +21 a^3 z-9 a^3 z^{-1} +5 a^2 z^8-7 a^2 z^6+7 z^6 a^{-2} -3 a^2 z^4-8 z^4 a^{-2} +11 a^2 z^2+z^2 a^{-2} +4 a^2 z^{-2} -10 a^2+a z^9+6 a z^7+7 z^7 a^{-1} -8 a z^5-8 z^5 a^{-1} -8 a z^3+13 a z+3 z a^{-1} -5 a z^{-1} - a^{-1} z^{-1} +4 z^8-z^6-3 z^4+z^2+ z^{-2} -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 \
9          11
7         3 -3
5        41 3
3       63  -3
1      54   1
-1     77    0
-3    44     0
-5   27      5
-7  34       -1
-9 15        4
-11           0
-131          1
Integral Khovanov Homology

(db, data source)

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