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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 L10a132's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,14,11,13 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X14,20,15,19 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 6, -5, 7, -4}, {10, -2, 3, -8, 4, -6, 5, -7, 8, -3}
A Braid Representative
A Morse Link Presentation L10a132 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+2 t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial  q^{-10} -2 q^{-9} +5 q^{-8} -7 q^{-7} +10 q^{-6} -9 q^{-5} +10 q^{-4} -7 q^{-3} +5 q^{-2} -3 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial a^{10} z^{-2} +a^{10}-3 a^8 z^2-2 a^8 z^{-2} -6 a^8+3 a^6 z^4+8 a^6 z^2+a^6 z^{-2} +5 a^6-a^4 z^6-3 a^4 z^4-2 a^4 z^2+a^2 z^4+2 a^2 z^2 (db)
Kauffman polynomial z^4 a^{12}-2 z^2 a^{12}+a^{12}+2 z^5 a^{11}-2 z^3 a^{11}+3 z^6 a^{10}-3 z^4 a^{10}+3 z^2 a^{10}+a^{10} z^{-2} -3 a^{10}+3 z^7 a^9-z^5 a^9-3 z^3 a^9+6 z a^9-2 a^9 z^{-1} +2 z^8 a^8+3 z^6 a^8-12 z^4 a^8+15 z^2 a^8+2 a^8 z^{-2} -8 a^8+z^9 a^7+3 z^7 a^7-5 z^5 a^7-3 z^3 a^7+6 z a^7-2 a^7 z^{-1} +5 z^8 a^6-9 z^6 a^6+6 z^2 a^6+a^6 z^{-2} -5 a^6+z^9 a^5+3 z^7 a^5-12 z^5 a^5+6 z^3 a^5+3 z^8 a^4-8 z^6 a^4+5 z^4 a^4-2 z^2 a^4+3 z^7 a^3-10 z^5 a^3+8 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^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 \
1          11
-1         2 -2
-3        31 2
-5       53  -2
-7      52   3
-9     45    1
-11    65     1
-13   25      3
-15  35       -2
-17 14        3
-19 1         -1
-211          1
Integral Khovanov Homology

(db, data source)

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