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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 L11n299's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,19,14,18 X17,11,18,22 X7,17,8,16 X21,8,22,9 X9,14,10,15 X15,20,16,21 X19,5,20,10 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -5, 6, -7, 9}, {11, -2, -3, 7, -8, 5, -4, 3, -9, 8, -6, 4}
A Braid Representative
A Morse Link Presentation L11n299 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(3)^4+t(2) t(3)^4+2 t(2)^2 t(3)^3-t(2) t(3)^3-t(3)^3-t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2+t(1) t(2)^2 t(3)-2 t(1) t(3)+t(1) t(2) t(3)+t(1)-t(1) t(2)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial - q^{-7} +3 q^{-6} -3 q^{-5} +4 q^{-4} +q^3-2 q^{-3} -2 q^2+3 q^{-2} +q (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6+a^6 z^{-2} -a^6+2 z^4 a^4+6 z^2 a^4-2 a^4 z^{-2} +a^4-z^6 a^2-5 z^4 a^2-6 z^2 a^2+a^2 z^{-2} -a^2+z^2+1+z^2 a^{-2} (db)
Kauffman polynomial 2 a^5 z^9+2 a^3 z^9+3 a^6 z^8+6 a^4 z^8+4 a^2 z^8+z^8+a^7 z^7-8 a^5 z^7-9 a^3 z^7+2 a z^7+2 z^7 a^{-1} -15 a^6 z^6-35 a^4 z^6-27 a^2 z^6+z^6 a^{-2} -6 z^6-4 a^7 z^5+a^5 z^5-15 a z^5-10 z^5 a^{-1} +19 a^6 z^4+55 a^4 z^4+48 a^2 z^4-4 z^4 a^{-2} +8 z^4+3 a^7 z^3+10 a^5 z^3+18 a^3 z^3+21 a z^3+10 z^3 a^{-1} -8 a^6 z^2-29 a^4 z^2-30 a^2 z^2+2 z^2 a^{-2} -7 z^2-a^7 z-3 a^5 z-7 a^3 z-7 a z-2 z a^{-1} +3 a^4+4 a^2+2-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 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 \
7           11
5          1 -1
3        111 -1
1       221  1
-1      231   0
-3     333    3
-5    351     1
-7   222      2
-9  241       1
-11 11         0
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

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