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

Visit L11n322 at Knotilus!

Link Presentations

[edit Notes on L11n322's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code {1, -4, 3, -7}, {-2, -1, 5, -3, -6, 9, -11, 10}, {-9, 2, 4, -5, 7, 6, -8, 11, -10, 8}
A Braid Representative
A Morse Link Presentation L11n322 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^2 (w-1)^2}{\sqrt{u} v w} (db)
Jones polynomial -2 q^3+5 q^2-7 q+11-10 q^{-1} +11 q^{-2} -8 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-4 a^2 z^4+3 z^4+2 a^4 z^2-8 a^2 z^2-2 z^2 a^{-2} +8 z^2+2 a^4-7 a^2-2 a^{-2} +7+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-9 a^5 z^5+7 a^5 z^3-a^5 z+3 a^4 z^8-5 a^4 z^6-4 a^4 z^4+5 a^4 z^2-a^4 z^{-2} +a^4+a^3 z^9+6 a^3 z^7-22 a^3 z^5+17 a^3 z^3+3 z^3 a^{-3} -6 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +6 a^2 z^8-11 a^2 z^6+z^6 a^{-2} +5 a^2 z^4+4 z^4 a^{-2} -8 a^2 z^2-5 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+3 a^{-2} +a z^9+6 a z^7+3 z^7 a^{-1} -17 a z^5-4 z^5 a^{-1} +17 a z^3+10 z^3 a^{-1} -10 a z-7 z a^{-1} +2 a z^{-1} +3 z^8-4 z^6+10 z^4-15 z^2- z^{-2} +9 (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 \
7         2-2
5        3 3
3       42 -2
1      73  4
-1     56   1
-3    65    1
-5   47     3
-7  24      -2
-9 14       3
-11 2        -2
-131         1
Integral Khovanov Homology

(db, data source)

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

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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