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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 L11a292's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,13,21,14 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,20,5,19
Gauss code {1, -2, 3, -11, 9, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 10, -9, 5, -4, 11, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L11a292 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial 3 q^{9/2}-\frac{3}{q^{9/2}}-6 q^{7/2}+\frac{7}{q^{7/2}}+10 q^{5/2}-\frac{12}{q^{5/2}}-14 q^{3/2}+\frac{14}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-z^5 a^{-3} -3 a^3 z^3-3 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a z^7+z^7 a^{-1} +4 a z^5+4 z^5 a^{-1} +6 a z^3+5 z^3 a^{-1} +4 a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -8 a^2 z^8+z^8 a^{-2} -3 z^8 a^{-4} -4 z^8-9 a^3 z^7+2 a z^7+27 z^7 a^{-1} +15 z^7 a^{-3} -z^7 a^{-5} -6 a^4 z^6+11 a^2 z^6+10 z^6 a^{-2} +12 z^6 a^{-4} +15 z^6-3 a^5 z^5+16 a^3 z^5+14 a z^5-28 z^5 a^{-1} -19 z^5 a^{-3} +4 z^5 a^{-5} -a^6 z^4+6 a^4 z^4-2 a^2 z^4-10 z^4 a^{-2} -14 z^4 a^{-4} -5 z^4+2 a^5 z^3-16 a^3 z^3-17 a z^3+17 z^3 a^{-1} +12 z^3 a^{-3} -4 z^3 a^{-5} +a^6 z^2-3 a^4 z^2-4 a^2 z^2+3 z^2 a^{-2} +4 z^2 a^{-4} -z^2+6 a^3 z+8 a z-2 z a^{-1} -4 z a^{-3} +1-a z^{-1} - a^{-1} 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 \
12           11
10          2 -2
8         41 3
6        62  -4
4       84   4
2      86    -2
0     98     1
-2    710      3
-4   57       -2
-6  27        5
-8 15         -4
-10 2          2
-121           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

Read me first: Modifying Knot Pages

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

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