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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 L10n92's Link Presentations]

Planar diagram presentation X8192 X18,10,19,9 X17,6,18,1 X7,17,8,16 X3,10,4,11 X5,14,6,15 X13,4,14,5 X11,13,12,20 X15,7,16,12 X2,19,3,20
Gauss code {1, -10, -5, 7, -6, 3}, {-4, -1, 2, 5, -8, 9}, {-7, 6, -9, 4, -3, -2, 10, 8}
A Braid Representative
A Morse Link Presentation L10n92 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^2 t(3)^3-t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2+t(1) t(2) t(3)^2-t(1)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)-t(1)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial  q^{-6} - q^{-5} +2 q^{-4} -2 q^{-3} +q^2+3 q^{-2} -q-2 q^{-1} +3 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+4 a^4 z^2+a^4 z^{-2} +4 a^4-a^2 z^6-6 a^2 z^4-12 a^2 z^2-2 a^2 z^{-2} -9 a^2+z^4+4 z^2+ z^{-2} +5 (db)
Kauffman polynomial a^6 z^6-5 a^6 z^4+6 a^6 z^2-a^6+a^5 z^7-4 a^5 z^5+3 a^5 z^3+a^4 z^8-5 a^4 z^6+9 a^4 z^4-10 a^4 z^2-a^4 z^{-2} +5 a^4+2 a^3 z^7-9 a^3 z^5+13 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+16 a^2 z^4-21 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +11 a^2- a^{-2} +a z^7-5 a z^5+11 a z^3+z^3 a^{-1} -9 a z+2 a z^{-1} +2 z^4-4 z^2- z^{-2} +5 (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       110
1      2  2
-1     23  1
-3    1    1
-5   12    1
-7  11     0
-9  1      1
-1111       0
-131        1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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