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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 L10a142's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X12,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X20,11,13,12 X10,19,11,20 X2536 X4,13,1,14
Gauss code {1, -9, 2, -10}, {9, -1, 5, -4, 6, -8, 7, -3}, {10, -2, 3, -5, 4, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L10a142 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^3 w^2+u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^2-2 u v w+u v+u w-u+v^3 w^3-v^3 w^2-v^2 w^3+2 v^2 w^2-v^2 w-v w^2+2 v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial  q^{-3} -2 q^{-4} +4 q^{-5} -4 q^{-6} +7 q^{-7} -7 q^{-8} +7 q^{-9} -5 q^{-10} +4 q^{-11} -2 q^{-12} + q^{-13} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{12} z^2+a^{12} z^{-2} +3 a^{12}-3 a^{10} z^4-12 a^{10} z^2-2 a^{10} z^{-2} -11 a^{10}+2 a^8 z^6+10 a^8 z^4+15 a^8 z^2+a^8 z^{-2} +8 a^8+a^6 z^6+4 a^6 z^4+3 a^6 z^2 (db)
Kauffman polynomial a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-3 a^{15} z^3+2 a^{14} z^6-a^{14} z^4-2 a^{14} z^2+2 a^{13} z^7-2 a^{13} z^5+a^{13} z^3+2 a^{12} z^8-5 a^{12} z^6+9 a^{12} z^4-8 a^{12} z^2-a^{12} z^{-2} +5 a^{12}+a^{11} z^9-8 a^{11} z^5+18 a^{11} z^3-11 a^{11} z+2 a^{11} z^{-1} +5 a^{10} z^8-22 a^{10} z^6+38 a^{10} z^4-31 a^{10} z^2-2 a^{10} z^{-2} +13 a^{10}+a^9 z^9-11 a^9 z^5+18 a^9 z^3-11 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-14 a^8 z^6+23 a^8 z^4-20 a^8 z^2-a^8 z^{-2} +8 a^8+2 a^7 z^7-7 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+3 a^6 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 \
-5          11
-7         21-1
-9        2  2
-11       22  0
-13      52   3
-15     33    0
-17    44     0
-19   13      2
-21  34       -1
-23 13        2
-25 1         -1
-271          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-7 i=-5
r=-10 {\mathbb Z}
r=-9 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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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