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

Visit L11n453 at Knotilus!

Link Presentations

[edit Notes on L11n453's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X15,2,16,3 X16,7,17,8 X19,22,20,15 X21,14,22,11 X13,20,14,21 X9,18,10,19 X11,10,12,5 X4,17,1,18
Gauss code {1, 4, -3, -11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {-4, -5, 11, 9, -6, 8, -7, 6}
A Braid Representative
A Morse Link Presentation L11n453 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v w^2 x^2-u v w^2 x-u v w x^2-u w^2 x^2+u w^2 x+u w x^2-u w x-v w x+v w+v x-v-w-x+2}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{6}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z a^{13}+a^{13} z^{-3} +z^5 a^{11}+4 z^3 a^{11}+z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} -z^7 a^9-4 z^5 a^9+11 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -z^7 a^7-6 z^5 a^7-12 z^3 a^7-11 z a^7-5 a^7 z^{-1} -a^7 z^{-3} (db)
Kauffman polynomial -z^2 a^{16}-3 z^3 a^{15}+3 z a^{15}-z^6 a^{14}+z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+12 z^5 a^{13}-18 z^3 a^{13}+12 z a^{13}-5 a^{13} z^{-1} +a^{13} z^{-3} -3 z^8 a^{12}+12 z^6 a^{12}-12 z^4 a^{12}-3 z^2 a^{12}-3 a^{12} z^{-2} +10 a^{12}-z^9 a^{11}+14 z^5 a^{11}-25 z^3 a^{11}+21 z a^{11}-12 a^{11} z^{-1} +3 a^{11} z^{-3} -4 z^8 a^{10}+16 z^6 a^{10}-10 z^4 a^{10}-15 z^2 a^{10}-6 a^{10} z^{-2} +19 a^{10}-z^9 a^9+2 z^7 a^9+8 z^5 a^9-22 z^3 a^9+23 z a^9-12 a^9 z^{-1} +3 a^9 z^{-3} -z^8 a^8+3 z^6 a^8+3 z^4 a^8-12 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^7 a^7+6 z^5 a^7-12 z^3 a^7+11 z a^7-5 a^7 z^{-1} +a^7 z^{-3} (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 \
-6         11
-8        110
-10       3  3
-12     111  1
-14     63   3
-16   222    2
-18   64     2
-20 124      3
-22 22       0
-24 2        2
-261         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{6} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6} {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\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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