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

Visit L11n39 at Knotilus!

Link Presentations

[edit Notes on L11n39's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X5,12,6,13 X8493 X9,16,10,17 X13,19,14,18 X17,15,18,14 X15,10,16,11 X11,22,12,5 X2,20,3,19
Gauss code {1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, 11, -2, 3, 10}
A Braid Representative
A Morse Link Presentation L11n39 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}} (db)
Jones polynomial -\frac{3}{q^{9/2}}+\frac{2}{q^{7/2}}-q^{5/2}-\frac{2}{q^{5/2}}+2 q^{3/2}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}-2 \sqrt{q} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-8 a^3 z-3 a^3 z^{-1} +a z^5+5 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^7-8 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +a^6 z^8-2 a^6 z^6-5 a^6 z^4+5 a^6 z^2-2 a^6+3 a^5 z^7-15 a^5 z^5+19 a^5 z^3-12 a^5 z+3 a^5 z^{-1} +2 a^4 z^8-12 a^4 z^6+17 a^4 z^4-7 a^4 z^2+a^3 z^9-4 a^3 z^7-7 a^3 z^5+27 a^3 z^3-18 a^3 z+3 a^3 z^{-1} +3 a^2 z^8-20 a^2 z^6+32 a^2 z^4-13 a^2 z^2+2 a^2+a z^9-4 a z^7+z^7 a^{-1} -5 a z^5-5 z^5 a^{-1} +21 a z^3+6 z^3 a^{-1} -13 a z-4 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +2 z^8-11 z^6+14 z^4-5 z^2 (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 \
6           11
4          1 -1
2         11 0
0       321  2
-2      231   0
-4     222    2
-6    231     0
-8   221      1
-10  121       0
-12 12         -1
-14 1          1
-161           -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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