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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n108 at Knotilus!

Link Presentations

[edit Notes on L10n108's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,11,20,14 X13,15,14,20 X9,18,10,19 X11,10,12,5 X4,17,1,18
Gauss code {1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {-4, -5, 10, 8, -6, 7}
A Braid Representative
A Morse Link Presentation L10n108 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v w^2 x^2-u v w^2 x-u w^2 x^2+u w x^2+v w-v-x+1}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial z a^9+a^9 z^{-1} +a^9 z^{-3} -z^5 a^7-5 z^3 a^7-7 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +z^7 a^5+6 z^5 a^5+12 z^3 a^5+13 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} -z^5 a^3-5 z^3 a^3-7 z a^3-4 a^3 z^{-1} -a^3 z^{-3} (db)
Kauffman polynomial a^{11} z+a^{10} z^2+3 a^9 z^3-a^9 z^{-3} -8 a^9 z+5 a^9 z^{-1} +2 a^8 z^6-9 a^8 z^4+13 a^8 z^2+3 a^8 z^{-2} -10 a^8+3 a^7 z^7-16 a^7 z^5+28 a^7 z^3-3 a^7 z^{-3} -25 a^7 z+12 a^7 z^{-1} +a^6 z^8-2 a^6 z^6-9 a^6 z^4+23 a^6 z^2+6 a^6 z^{-2} -19 a^6+4 a^5 z^7-22 a^5 z^5+37 a^5 z^3-3 a^5 z^{-3} -27 a^5 z+12 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+11 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-a^3 z^{-3} -11 a^3 z+5 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 \
0        11
-2         0
-4      31 2
-6    112  2
-8    41   3
-10  212    3
-12  42     2
-141 1      2
-1622       0
-181        1
Integral Khovanov Homology

(db, data source)

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