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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 L10a171's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X10,13,5,14 X20,16,17,15 X14,7,15,8 X16,20,11,19 X8,17,9,18 X18,9,19,10 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {7, -8, 6, -4}, {9, -1, 5, -7, 8, -3}, {10, -2, 3, -5, 4, -6}
A Braid Representative
A Morse Link Presentation L10a171 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^2 x-u v w^2+u v w x^2-3 u v w x+2 u v w+u v x-u v-u w^2 x-u w x^2+3 u w x-u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-v w x^2+3 v w x-v w-v x-w^2 x^2+w^2 x+2 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^9+a^9 z^{-1} +a^9 z^{-3} -3 z^3 a^7-6 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +2 z^5 a^5+6 z^3 a^5+10 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-4 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a-z a (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-10 z^3 a^9+11 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -4 z^8 a^8+2 z^6 a^8+9 z^4 a^8-14 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^9 a^7-13 z^7 a^7+42 z^5 a^7-54 z^3 a^7+33 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -8 z^8 a^6+10 z^6 a^6+8 z^4 a^6-26 z^2 a^6-6 a^6 z^{-2} +19 a^6-z^9 a^5-13 z^7 a^5+42 z^5 a^5-54 z^3 a^5+33 z a^5-12 a^5 z^{-1} +3 a^5 z^{-3} -4 z^8 a^4+2 z^6 a^4+9 z^4 a^4-14 z^2 a^4-3 a^4 z^{-2} +10 a^4-5 z^7 a^3+10 z^5 a^3-10 z^3 a^3+11 z a^3-5 a^3 z^{-1} +a^3 z^{-3} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a (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 \
2          11
0         2 -2
-2        51 4
-4       64  -2
-6      83   5
-8     68    2
-10    86     2
-12   38      5
-14  46       -2
-16 15        4
-18 2         -2
-201          1
Integral Khovanov Homology

(db, data source)

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