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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 L10a153's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,11,19,12 X8,17,9,18 X16,7,17,8 X20,13,15,14 X14,15,5,16 X12,19,13,20 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {7, -5, 4, -3, 8, -6}, {9, -1, 5, -4, 10, -2, 3, -8, 6, -7}
A Braid Representative
A Morse Link Presentation L10a153 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(2 u v^2 w^2-u v w^2+2 u v w-u w+u+v^2 w^3-v^2 w^2+2 v w^2-v w+2 w\right)}{\sqrt{u} v w^2} (db)
Jones polynomial  q^{-3} -2 q^{-4} +5 q^{-5} -6 q^{-6} +9 q^{-7} -9 q^{-8} +9 q^{-9} -7 q^{-10} +5 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^4-11 a^{10} z^2-2 a^{10} z^{-2} -10 a^{10}+2 a^8 z^6+9 a^8 z^4+12 a^8 z^2+a^8 z^{-2} +6 a^8+a^6 z^6+4 a^6 z^4+4 a^6 z^2+a^6 (db)
Kauffman polynomial a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-2 a^{15} z^3+3 a^{14} z^6-3 a^{14} z^4+a^{14} z^2+3 a^{13} z^7-2 a^{13} z^5+3 a^{12} z^8-6 a^{12} z^6+11 a^{12} z^4-11 a^{12} z^2-a^{12} z^{-2} +6 a^{12}+a^{11} z^9+4 a^{11} z^7-17 a^{11} z^5+24 a^{11} z^3-13 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-21 a^{10} z^6+34 a^{10} z^4-32 a^{10} z^2-2 a^{10} z^{-2} +14 a^{10}+a^9 z^9+3 a^9 z^7-19 a^9 z^5+25 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+15 a^8 z^4-14 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+3 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^2-a^6 (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        3  3
-11       32  -1
-13      63   3
-15     44    0
-17    55     0
-19   35      2
-21  24       -2
-23  3        3
-2512         -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} {\mathbb Z}
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\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=-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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