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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 L11n363's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X5,18,6,19 X8493 X9,21,10,20 X19,11,20,10 X15,22,16,17 X17,16,18,5 X21,14,22,15 X2,12,3,11
Gauss code {1, -11, 5, -3}, {-9, 4, -7, 6, -10, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10, -8, 9}
A Braid Representative
A Morse Link Presentation L11n363 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(3)-1)^2}{\sqrt{t(1)} t(3)} (db)
Jones polynomial - q^{-8} +2 q^{-7} -3 q^{-6} +4 q^{-5} -3 q^{-4} +4 q^{-3} +q^2- q^{-2} -q+ q^{-1} +1 (db)
Signature -2 (db)
HOMFLY-PT polynomial -z^4 a^6-3 z^2 a^6-3 a^6+z^6 a^4+6 z^4 a^4+13 z^2 a^4+a^4 z^{-2} +9 a^4-z^6 a^2-7 z^4 a^2-14 z^2 a^2-2 a^2 z^{-2} -10 a^2+z^4+4 z^2+ z^{-2} +4 (db)
Kauffman polynomial z^5 a^9-3 z^3 a^9+z a^9+2 z^6 a^8-6 z^4 a^8+3 z^2 a^8-a^8+2 z^7 a^7-6 z^5 a^7+3 z^3 a^7+z^8 a^6-3 z^6 a^6+3 z^4 a^6-3 z^2 a^6+2 a^6+2 z^7 a^5-9 z^5 a^5+14 z^3 a^5-6 z a^5+2 z^8 a^4-15 z^6 a^4+37 z^4 a^4-34 z^2 a^4-a^4 z^{-2} +13 a^4+z^9 a^3-7 z^7 a^3+11 z^5 a^3+3 z^3 a^3-9 z a^3+2 a^3 z^{-1} +2 z^8 a^2-17 z^6 a^2+43 z^4 a^2-41 z^2 a^2-2 a^2 z^{-2} +15 a^2+z^9 a-7 z^7 a+13 z^5 a-5 z^3 a-4 z a+2 a z^{-1} +z^8-7 z^6+15 z^4-13 z^2- z^{-2} +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
3            0
1         11 0
-1       31   2
-3      121   0
-5     421    3
-7    23      1
-9   221      1
-11  12        1
-13 12         -1
-15 1          1
-171           -1
Integral Khovanov Homology

(db, data source)

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