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

Visit L11n178 at Knotilus!

Link Presentations

[edit Notes on L11n178's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X13,21,14,20 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,22,16,7 X11,4,12,5 X19,13,20,12 X21,1,22,6 X2,18,3,17
Gauss code {1, -11, -4, 8, -5, 10}, {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}
A Braid Representative
A Morse Link Presentation L11n178 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u v^2} (db)
Jones polynomial -q^{3/2}+3 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7-a^5 z^5+5 a^3 z^5-a z^5-3 a^5 z^3+8 a^3 z^3-3 a z^3-2 a^5 z+4 a^3 z-3 a z+a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-3 a^6 z^8-4 a^4 z^8-a^2 z^8-3 a^7 z^7-2 a^5 z^7+a^3 z^7-a^8 z^6+8 a^6 z^6+8 a^4 z^6-a^2 z^6+10 a^7 z^5+10 a^5 z^5-6 a^3 z^5-6 a z^5+3 a^8 z^4-3 a^6 z^4-3 a^4 z^4-3 z^4-8 a^7 z^3-4 a^5 z^3+12 a^3 z^3+7 a z^3-z^3 a^{-1} -2 a^8 z^2-a^6 z^2+a^4 z^2+a^2 z^2+z^2+a^7 z-a^5 z-6 a^3 z-4 a z-a^2+a^3 z^{-1} +a 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 \
4         11
2        2 -2
0       41 3
-2      43  -1
-4     53   2
-6    45    1
-8   34     -1
-10  24      2
-12 13       -2
-14 2        2
-161         -1
Integral Khovanov Homology

(db, data source)

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