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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 L11a345's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,15,21,16 X14,21,15,22 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code {1, -11, 2, -10, 9, -7, 8, -2, 10, -9}, {11, -1, 3, -6, 5, -8, 7, -3, 4, -5, 6, -4}
A Braid Representative
A Morse Link Presentation L11a345 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^3-4 u^3 v^2+6 u^3 v-2 u^3-2 u^2 v^3+6 u^2 v^2-6 u^2 v+2 u^2+2 u v^3-6 u v^2+6 u v-2 u-2 v^3+6 v^2-4 v+1}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-19 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+10 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-8 a^3 z+12 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -5 a^3 z^{-1} +6 a z^{-1} -4 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +a^5 z^7-5 a^5 z^5+4 z^5 a^{-5} +9 a^5 z^3-z^3 a^{-5} -7 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-7 a^4 z^6+9 z^6 a^{-4} +7 a^4 z^4-8 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} -a^4- a^{-4} +2 a^3 z^9-a^3 z^7+13 z^7 a^{-3} -16 a^3 z^5-19 z^5 a^{-3} +31 a^3 z^3+11 z^3 a^{-3} -21 a^3 z-4 z a^{-3} +5 a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^{10}+6 a^2 z^8+11 z^8 a^{-2} -24 a^2 z^6-12 z^6 a^{-2} +21 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+9 z^2 a^{-2} -a^2-3 a^{-2} +7 a z^9+5 z^9 a^{-1} -6 a z^7+9 z^7 a^{-1} -30 a z^5-42 z^5 a^{-1} +50 a z^3+40 z^3 a^{-1} -29 a z-19 z a^{-1} +6 a z^{-1} +4 a^{-1} z^{-1} +z^{10}+15 z^8-38 z^6+18 z^4+4 z^2-3 (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           1-1
10          3 3
8         61 -5
6        83  5
4       96   -3
2      108    2
0     910     1
-2    69      -3
-4   49       5
-6  26        -4
-8 15         4
-10 1          -1
-121           1
Integral Khovanov Homology

(db, data source)

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