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

Visit L11n211 at Knotilus!

Link Presentations

[edit Notes on L11n211's Link Presentations]

Planar diagram presentation X10,1,11,2 X15,21,16,20 X5,14,6,15 X3,12,4,13 X13,4,14,5 X2,19,3,20 X16,7,17,8 X8,9,1,10 X18,12,19,11 X22,18,9,17 X21,6,22,7
Gauss code {1, -6, -4, 5, -3, 11, 7, -8}, {8, -1, 9, 4, -5, 3, -2, -7, 10, -9, 6, 2, -11, -10}
A Braid Representative
A Morse Link Presentation L11n211 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^2+u^2 v^5-2 u^2 v^4+3 u^2 v^3-3 u^2 v^2+2 u^2 v+2 u v^4-3 u v^3+3 u v^2-2 u v+u+v^3}{u^{3/2} v^{5/2}} (db)
Jones polynomial \frac{8}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{11/2}}+\frac{1}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+4 z^3 a^7+6 z a^7+2 a^7 z^{-1} -z^7 a^5-6 z^5 a^5-14 z^3 a^5-13 z a^5-3 a^5 z^{-1} +z^5 a^3+3 z^3 a^3+3 z a^3+a^3 z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-3 z^7 a^9+7 z^5 a^9-6 z^3 a^9+3 z a^9-2 z^8 a^8+2 z^6 a^8+z^2 a^8-z^9 a^7+z^7 a^7-6 z^5 a^7+11 z^3 a^7-8 z a^7+2 a^7 z^{-1} -3 z^8 a^6+7 z^6 a^6-14 z^4 a^6+12 z^2 a^6-3 a^6-z^9 a^5+4 z^7 a^5-17 z^5 a^5+25 z^3 a^5-16 z a^5+3 a^5 z^{-1} -z^8 a^4+3 z^6 a^4-11 z^4 a^4+12 z^2 a^4-3 a^4-3 z^5 a^3+5 z^3 a^3-3 z a^3+a^3 z^{-1} -z^4 a^2+2 z^2 a^2-a^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 \
0         1-1
-2        2 2
-4       42 -2
-6      41  3
-8     44   0
-10    44    0
-12   34     1
-14  24      -2
-16  3       3
-1812        -1
-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} {\mathbb Z}
r=-7 {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\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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