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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 L11a253's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,15,21,16 X16,8,17,7 X4,13,5,14 X6,20,7,19
Gauss code {1, -2, 3, -10, 4, -11, 9, -7}, {5, -1, 2, -3, 10, -4, 8, -9, 6, -5, 11, -8, 7, -6}
A Braid Representative
A Morse Link Presentation L11a253 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+3 t(1)^2 t(2)^4-t(1) t(2)^4+2 t(1)^3 t(2)^3-5 t(1)^2 t(2)^3+3 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \frac{10}{q^{9/2}}-\frac{12}{q^{7/2}}-q^{5/2}+\frac{11}{q^{5/2}}+3 q^{3/2}-\frac{10}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-5 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^3 z^9-a^5 z^7+7 a^3 z^7-a z^7-5 a^5 z^5+17 a^3 z^5-5 a z^5-7 a^5 z^3+16 a^3 z^3-7 a z^3-2 a^5 z+3 a^3 z-3 a z+a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+4 z^3 a^9-z a^9-4 z^6 a^8+4 z^4 a^8-4 z^7 a^7+3 z^5 a^7+2 z^3 a^7-2 z a^7-4 z^8 a^6+6 z^6 a^6-3 z^4 a^6-4 z^9 a^5+12 z^7 a^5-17 z^5 a^5+11 z^3 a^5-z a^5-a^5 z^{-1} -2 z^{10} a^4+3 z^8 a^4+5 z^6 a^4-10 z^4 a^4+3 z^2 a^4+a^4-8 z^9 a^3+35 z^7 a^3-51 z^5 a^3+30 z^3 a^3-4 z a^3-a^3 z^{-1} -2 z^{10} a^2+4 z^8 a^2+8 z^6 a^2-16 z^4 a^2+6 z^2 a^2-4 z^9 a+18 z^7 a-24 z^5 a+14 z^3 a-4 z a-3 z^8+13 z^6-14 z^4+4 z^2-z^7 a^{-1} +4 z^5 a^{-1} -3 z^3 a^{-1} (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 \
6           11
4          2 -2
2         31 2
0        42  -2
-2       63   3
-4      65    -1
-6     65     1
-8    46      2
-10   46       -2
-12  25        3
-14 13         -2
-16 2          2
-181           -1
Integral Khovanov Homology

(db, data source)

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

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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