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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 L11a331's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X22,18,9,17 X18,14,19,13 X14,22,15,21 X16,7,17,8 X8,9,1,10 X6,15,7,16 X4,19,5,20
Gauss code {1, -2, 3, -11, 4, -10, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -8, 5, -6, 11, -4, 7, -5}
A Braid Representative
A Morse Link Presentation L11a331 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^3 t(2)^5-3 t(1)^3 t(2)^4+3 t(1)^2 t(2)^4+3 t(1)^3 t(2)^3-6 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-6 t(1) t(2)^2+3 t(2)^2+3 t(1) t(2)-3 t(2)+1}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^7 z^7+5 a^7 z^5+8 a^7 z^3+6 a^7 z+2 a^7 z^{-1} -a^5 z^9-7 a^5 z^7-18 a^5 z^5-22 a^5 z^3-13 a^5 z-3 a^5 z^{-1} +a^3 z^7+5 a^3 z^5+7 a^3 z^3+3 a^3 z+a^3 z^{-1} (db)
Kauffman polynomial -z^3 a^{13}-3 z^4 a^{12}-6 z^5 a^{11}+5 z^3 a^{11}-2 z a^{11}-8 z^6 a^{10}+10 z^4 a^{10}-2 z^2 a^{10}-9 z^7 a^9+17 z^5 a^9-7 z^3 a^9+3 z a^9-8 z^8 a^8+18 z^6 a^8-6 z^4 a^8+z^2 a^8-6 z^9 a^7+17 z^7 a^7-12 z^5 a^7+10 z^3 a^7-8 z a^7+2 a^7 z^{-1} -2 z^{10} a^6-2 z^8 a^6+31 z^6 a^6-43 z^4 a^6+18 z^2 a^6-3 a^6-9 z^9 a^5+42 z^7 a^5-62 z^5 a^5+39 z^3 a^5-16 z a^5+3 a^5 z^{-1} -2 z^{10} a^4+5 z^8 a^4+10 z^6 a^4-32 z^4 a^4+20 z^2 a^4-3 a^4-3 z^9 a^3+16 z^7 a^3-27 z^5 a^3+16 z^3 a^3-3 z a^3+a^3 z^{-1} -z^8 a^2+5 z^6 a^2-8 z^4 a^2+5 z^2 a^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 \
2           1-1
0          2 2
-2         31 -2
-4        52  3
-6       64   -2
-8      64    2
-10     66     0
-12    56      -1
-14   36       3
-16  35        -2
-18  3         3
-2013          -2
-221           1
Integral Khovanov Homology

(db, data source)

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

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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