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

Visit L11n155 at Knotilus!

Link Presentations

[edit Notes on L11n155's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X7,16,8,17 X13,20,14,21 X15,22,16,7 X6,19,1,20 X18,11,19,12 X12,6,13,5 X21,14,22,15 X4,18,5,17
Gauss code {1, -2, 3, -11, 9, -7}, {-4, -1, 2, -3, 8, -9, -5, 10, -6, 4, 11, -8, 7, 5, -10, 6}
A Braid Representative
A Morse Link Presentation L11n155 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4-2 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -\frac{2}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{8}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-3 z^3 a^9-3 z a^9-2 a^9 z^{-1} +z^7 a^7+5 z^5 a^7+10 z^3 a^7+11 z a^7+5 a^7 z^{-1} -2 z^5 a^5-8 z^3 a^5-9 z a^5-3 a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+5 a^{12} z^6-7 a^{12} z^4+4 a^{12} z^2-a^{12}+5 a^{11} z^7-7 a^{11} z^5+5 a^{11} z^3-a^{11} z+3 a^{10} z^8-a^{10} z^6-3 a^{10} z^4+3 a^{10} z^2+a^9 z^9+3 a^9 z^7-5 a^9 z^5-a^9 z^3+4 a^9 z-2 a^9 z^{-1} +4 a^8 z^8-8 a^8 z^6+9 a^8 z^4-11 a^8 z^2+5 a^8+a^7 z^9-2 a^7 z^7+8 a^7 z^5-19 a^7 z^3+15 a^7 z-5 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+4 a^6 z^4-9 a^6 z^2+5 a^6+3 a^5 z^5-10 a^5 z^3+10 a^5 z-3 a^5 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 \
-4         22
-6        21-1
-8       51 4
-10      43  -1
-12     64   2
-14    44    0
-16   46     -2
-18  24      2
-20 14       -3
-22 2        2
-241         -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}

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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