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

Visit L11n213 at Knotilus!

Link Presentations

[edit Notes on L11n213's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X9,18,10,19 X17,22,18,9 X21,1,22,8 X20,15,21,16 X7,16,8,17 X4,13,5,14 X6,20,7,19
Gauss code {1, -2, 3, -10, 4, -11, -9, 7}, {-5, -1, 2, -3, 10, -4, 8, 9, -6, 5, 11, -8, -7, 6}
A Braid Representative
A Morse Link Presentation L11n213 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^9 \left(-z^5\right)-4 a^9 z^3-4 a^9 z-a^9 z^{-1} +a^7 z^7+6 a^7 z^5+13 a^7 z^3+12 a^7 z+3 a^7 z^{-1} -2 a^5 z^5-9 a^5 z^3-10 a^5 z-2 a^5 z^{-1} (db)
Kauffman polynomial a^{14} z^4-2 a^{14} z^2+2 a^{13} z^5-4 a^{13} z^3+a^{13} z+2 a^{12} z^6-3 a^{12} z^4+a^{12} z^2+2 a^{11} z^7-4 a^{11} z^5+4 a^{11} z^3+2 a^{10} z^8-7 a^{10} z^6+12 a^{10} z^4-5 a^{10} z^2+a^{10}+a^9 z^9-3 a^9 z^7+5 a^9 z^5-3 a^9 z^3+3 a^9 z-a^9 z^{-1} +3 a^8 z^8-13 a^8 z^6+23 a^8 z^4-17 a^8 z^2+3 a^8+a^7 z^9-5 a^7 z^7+14 a^7 z^5-23 a^7 z^3+15 a^7 z-3 a^7 z^{-1} +a^6 z^8-4 a^6 z^6+7 a^6 z^4-9 a^6 z^2+3 a^6+3 a^5 z^5-12 a^5 z^3+11 a^5 z-2 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        110
-8       31 2
-10      21  -1
-12     43   1
-14    33    0
-16   23     -1
-18  13      2
-20 12       -1
-22 1        1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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See/edit the Link Page master template (intermediate).

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