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

Visit L11n169 at Knotilus!

Link Presentations

[edit Notes on L11n169's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X15,20,16,21 X11,18,12,19 X19,12,20,13 X17,22,18,7 X21,16,22,17 X6718 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -6, 7, 11, -4, -5, 9, -8, 6, -7, 5, -9, 8}
A Braid Representative
A Morse Link Presentation L11n169 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^6-u^2 v^4+u^2 v^3+2 u v^4-3 u v^3+2 u v^2+v^3-v^2+1}{u v^3} (db)
Jones polynomial -\frac{1}{q^{9/2}}+\frac{1}{q^{29/2}}-\frac{2}{q^{27/2}}+\frac{3}{q^{25/2}}-\frac{4}{q^{23/2}}+\frac{4}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -2 z^3 a^{13}-5 z a^{13}-2 a^{13} z^{-1} +z^7 a^{11}+9 z^5 a^{11}+24 z^3 a^{11}+22 z a^{11}+5 a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-37 z^3 a^9-20 z a^9-3 a^9 z^{-1} (db)
Kauffman polynomial a^{18} z^4-2 a^{18} z^2+2 a^{17} z^5-4 a^{17} z^3+a^{17} z+2 a^{16} z^6-4 a^{16} z^4+3 a^{16} z^2-a^{16}+a^{15} z^7-a^{15} z^5+a^{15} z^3+2 a^{14} z^6-3 a^{14} z^4+3 a^{14} z^2+2 a^{13} z^5-3 a^{13} z^3+5 a^{13} z-2 a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-24 a^{12} z^2+5 a^{12}+a^{11} z^9-10 a^{11} z^7+33 a^{11} z^5-45 a^{11} z^3+26 a^{11} z-5 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+24 a^{10} z^4-22 a^{10} z^2+5 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+20 a^9 z-3 a^9 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 \
-8           11
-10           11
-12         1  1
-14       2    2
-16      111   -1
-18     32     1
-20    221     -1
-22   22       0
-24  12        1
-26 12         -1
-28 1          1
-301           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-10 i=-8 i=-6
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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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