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

Visit L11n210 at Knotilus!

Link Presentations

[edit Notes on L11n210's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X22,18,9,17 X4,19,5,20 X21,6,22,7 X16,7,17,8 X8,9,1,10 X18,14,19,13 X15,21,16,20
Gauss code {1, -2, 3, -6, -4, 7, 8, -9}, {9, -1, 2, -3, 10, 4, -11, -8, 5, -10, 6, 11, -7, -5}
A Braid Representative
A Morse Link Presentation L11n210 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^2+u^2 v^5-u^2 v^4+u^2 v^3-u^2 v^2+u^2 v+u v^4-u v^3+u v^2-u v+u+v^3}{u^{3/2} v^{5/2}} (db)
Jones polynomial \frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+5 z^3 a^7+7 z a^7+2 a^7 z^{-1} -z^7 a^5-7 z^5 a^5-17 z^3 a^5-15 z a^5-3 a^5 z^{-1} +z^5 a^3+4 z^3 a^3+4 z a^3+a^3 z^{-1} (db)
Kauffman polynomial a^{11} z^5-4 a^{11} z^3+3 a^{11} z+a^{10} z^6-3 a^{10} z^4+a^{10} z^2+a^9 z^7-3 a^9 z^5+2 a^9 z^3-a^9 z+a^8 z^8-4 a^8 z^6+6 a^8 z^4-3 a^8 z^2+a^7 z^9-6 a^7 z^7+15 a^7 z^5-16 a^7 z^3+10 a^7 z-2 a^7 z^{-1} +2 a^6 z^8-11 a^6 z^6+23 a^6 z^4-15 a^6 z^2+3 a^6+a^5 z^9-7 a^5 z^7+21 a^5 z^5-28 a^5 z^3+18 a^5 z-3 a^5 z^{-1} +a^4 z^8-6 a^4 z^6+15 a^4 z^4-14 a^4 z^2+3 a^4+2 a^3 z^5-6 a^3 z^3+4 a^3 z-a^3 z^{-1} +a^2 z^4-3 a^2 z^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        1 1
-4       22 0
-6      2   2
-8     22   0
-10    22    0
-12   12     1
-14  12      -1
-16  1       1
-1811        0
-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}
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}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2} {\mathbb Z}
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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