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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 L11n339's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X7,14,8,15 X13,8,14,9 X19,22,20,13 X15,20,16,21 X21,16,22,17 X12,18,5,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, -7, 8, 9, -3, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation L11n339 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-1) (t(3)-1) \left(t(3) t(2)^2+t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-t(2)-t(1) t(3)\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial 1-2 q^{-1} +4 q^{-2} -3 q^{-3} +4 q^{-4} -2 q^{-5} +3 q^{-6} - q^{-8} + q^{-9} - q^{-10} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} -a^{10}+2 z^2 a^8+4 a^8 z^{-2} +5 a^8-2 z^2 a^6-5 a^6 z^{-2} -7 a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4+2 a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+2 a^2 (db)
Kauffman polynomial z^7 a^{11}-6 z^5 a^{11}+10 z^3 a^{11}-6 z a^{11}+a^{11} z^{-1} +z^8 a^{10}-6 z^6 a^{10}+9 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +4 a^{10}+2 z^7 a^9-16 z^5 a^9+33 z^3 a^9-22 z a^9+5 a^9 z^{-1} +z^8 a^8-9 z^6 a^8+22 z^4 a^8-22 z^2 a^8-4 a^8 z^{-2} +15 a^8+2 z^7 a^7-15 z^5 a^7+34 z^3 a^7-30 z a^7+9 a^7 z^{-1} +z^8 a^6-5 z^6 a^6+11 z^4 a^6-19 z^2 a^6-5 a^6 z^{-2} +16 a^6+3 z^7 a^5-12 z^5 a^5+15 z^3 a^5-13 z a^5+5 a^5 z^{-1} +z^8 a^4-z^6 a^4-6 z^4 a^4+2 z^2 a^4-2 a^4 z^{-2} +4 a^4+2 z^7 a^3-7 z^5 a^3+4 z^3 a^3+z a^3+z^6 a^2-4 z^4 a^2+5 z^2 a^2-2 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          1 -1
-3         31 2
-5        23  1
-7      131   1
-9     112    2
-11    153     1
-13   1 2      3
-15   12       -1
-17 11         0
-19            0
-211           -1
Integral Khovanov Homology

(db, data source)

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