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

Visit L11n7 at Knotilus!

Link Presentations

[edit Notes on L11n7's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X15,2,16,3
Gauss code {1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, -8, 7, -10, 8}
A Braid Representative
A Morse Link Presentation L11n7 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-4 t(1) t(2)^4+4 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+4 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{5}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +z^3 a^9+4 z a^9+3 a^9 z^{-1} -2 z^5 a^7-8 z^3 a^7-8 z a^7-3 a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+6 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1} (db)
Kauffman polynomial -3 z^2 a^{12}+a^{12}-z^5 a^{11}-3 z^3 a^{11}+3 z a^{11}-a^{11} z^{-1} -3 z^6 a^{10}+5 z^4 a^{10}-6 z^2 a^{10}+2 a^{10}-4 z^7 a^9+11 z^5 a^9-15 z^3 a^9+12 z a^9-3 a^9 z^{-1} -3 z^8 a^8+7 z^6 a^8-4 z^4 a^8+3 z^2 a^8-z^9 a^7-3 z^7 a^7+21 z^5 a^7-27 z^3 a^7+15 z a^7-3 a^7 z^{-1} -5 z^8 a^6+19 z^6 a^6-20 z^4 a^6+9 z^2 a^6-2 a^6-z^9 a^5+14 z^5 a^5-23 z^3 a^5+11 z a^5-2 a^5 z^{-1} -2 z^8 a^4+9 z^6 a^4-11 z^4 a^4+3 z^2 a^4-z^7 a^3+5 z^5 a^3-8 z^3 a^3+5 z a^3-a^3 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 \
0         11
-2        1 -1
-4       31 2
-6      32  -1
-8     42   2
-10    33    0
-12   44     0
-14  24      2
-16 23       -1
-18 2        2
-202         -2
Integral Khovanov Homology

(db, data source)

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