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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 L11a327's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,10,19,9 X6,13,7,14 X14,7,15,8 X8,15,1,16 X22,18,9,17 X16,22,17,21 X4,20,5,19 X20,6,21,5
Gauss code {1, -2, 3, -10, 11, -5, 6, -7}, {4, -1, 2, -3, 5, -6, 7, -9, 8, -4, 10, -11, 9, -8}
A Braid Representative
A Morse Link Presentation L11a327 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u^3 v^5+u^3 v^4-u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+5 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+5 u v^2-4 u v+u+v^3-v^2+v-1}{u^{3/2} v^{5/2}} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+5 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+6 a^3 z^5+13 a^3 z^3+10 a^3 z+2 a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -25 a z^5+6 z^5 a^{-1} -36 a z^3+13 z^3 a^{-1} -22 a z+10 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+2 a^7 z+2 a^6 z^6-4 a^6 z^4+a^6 z^2+3 a^5 z^7-5 a^5 z^5+2 a^5 z^3-a^5 z+4 a^4 z^8-10 a^4 z^6+z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -7 a^4 z^2+4 z^2 a^{-4} +3 a^3 z^9-6 a^3 z^7+2 z^7 a^{-3} +10 a^3 z^5-6 z^5 a^{-3} -10 a^3 z^3+4 z^3 a^{-3} +9 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+4 a^2 z^8+3 z^8 a^{-2} -17 a^2 z^6-9 z^6 a^{-2} +30 a^2 z^4+10 z^4 a^{-2} -18 a^2 z^2-7 z^2 a^{-2} +3 a^2+ a^{-2} +6 a z^9+3 z^9 a^{-1} -21 a z^7-10 z^7 a^{-1} +43 a z^5+21 z^5 a^{-1} -47 a z^3-28 z^3 a^{-1} +24 a z+12 z a^{-1} -3 a z^{-1} - a^{-1} z^{-1} +z^{10}+3 z^8-15 z^6+26 z^4-21 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 \
10           1-1
8          1 1
6         41 -3
4        51  4
2       64   -2
0      85    3
-2     67     1
-4    67      -1
-6   36       3
-8  26        -4
-10 14         3
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

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