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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 L11a358's Link Presentations]

Planar diagram presentation X12,1,13,2 X14,3,15,4 X20,10,21,9 X16,6,17,5 X18,8,19,7 X22,15,11,16 X6,18,7,17 X8,20,9,19 X4,22,5,21 X2,11,3,12 X10,13,1,14
Gauss code {1, -10, 2, -9, 4, -7, 5, -8, 3, -11}, {10, -1, 11, -2, 6, -4, 7, -5, 8, -3, 9, -6}
A Braid Representative
A Morse Link Presentation L11a358 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^3 t(1)^4-t(2)^2 t(1)^4+t(2)^4 t(1)^3-3 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)-t(2)^2+t(2)}{t(1)^2 t(2)^2} (db)
Jones polynomial -q^{15/2}+3 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-1} +z^7 a^{-3} -a z^5+5 z^5 a^{-1} +4 z^5 a^{-3} -z^5 a^{-5} -4 a z^3+8 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} -3 a z+6 z a^{-1} -z a^{-3} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10}-2 a z^9-5 z^9 a^{-1} -3 z^9 a^{-3} -a^2 z^8-2 z^8 a^{-2} -5 z^8 a^{-4} +2 z^8+12 a z^7+24 z^7 a^{-1} +6 z^7 a^{-3} -6 z^7 a^{-5} +6 a^2 z^6+19 z^6 a^{-2} +11 z^6 a^{-4} -6 z^6 a^{-6} +8 z^6-24 a z^5-37 z^5 a^{-1} +2 z^5 a^{-3} +10 z^5 a^{-5} -5 z^5 a^{-7} -11 a^2 z^4-23 z^4 a^{-2} -4 z^4 a^{-4} +7 z^4 a^{-6} -3 z^4 a^{-8} -20 z^4+19 a z^3+25 z^3 a^{-1} -z^3 a^{-3} -2 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +6 a^2 z^2+7 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +10 z^2-5 a z-9 z a^{-1} -3 z a^{-3} -z a^{-7} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-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 \
16           11
14          2 -2
12         31 2
10        42  -2
8       53   2
6      55    0
4     44     0
2    46      2
0   23       -1
-2  14        3
-4 12         -1
-6 1          1
-81           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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See/edit the Link Page master template (intermediate).

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