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

Visit L11a286 at Knotilus!

Link Presentations

[edit Notes on L11a286's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,8,19,7 X16,9,17,10 X22,15,9,16 X4,21,5,22 X14,6,15,5 X20,14,21,13 X8,18,1,17 X6,20,7,19
Gauss code {1, -2, 3, -7, 8, -11, 4, -10}, {5, -1, 2, -3, 9, -8, 6, -5, 10, -4, 11, -9, 7, -6}
A Braid Representative
A Morse Link Presentation L11a286 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3-t(1)^2 t(2)^2-t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial q^{7/2}-3 q^{5/2}+5 q^{3/2}-9 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^3 z^7-a z^7+a^5 z^5-4 a^3 z^5-4 a z^5+z^5 a^{-1} +3 a^5 z^3-4 a^3 z^3-4 a z^3+3 z^3 a^{-1} +a^5 z-a^3 z-a z+z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^9 z^3+3 a^8 z^4-a^8 z^2+5 a^7 z^5-3 a^7 z^3+7 a^6 z^6-8 a^6 z^4+9 a^5 z^7-19 a^5 z^5+9 a^5 z^3-2 a^5 z+9 a^4 z^8-25 a^4 z^6+17 a^4 z^4-4 a^4 z^2+6 a^3 z^9-15 a^3 z^7+9 a^3 z^3-2 a^3 z+2 a^2 z^{10}+3 a^2 z^8+z^8 a^{-2} -35 a^2 z^6-5 z^6 a^{-2} +44 a^2 z^4+8 z^4 a^{-2} -12 a^2 z^2-4 z^2 a^{-2} +9 a z^9+3 z^9 a^{-1} -40 a z^7-16 z^7 a^{-1} +52 a z^5+28 z^5 a^{-1} -22 a z^3-18 z^3 a^{-1} +2 a z+2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}-5 z^8-8 z^6+24 z^4-11 z^2-1 (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           1-1
6          2 2
4         31 -2
2        62  4
0       43   -1
-2      86    2
-4     66     0
-6    56      -1
-8   36       3
-10  25        -3
-12 13         2
-14 2          -2
-161           1
Integral Khovanov Homology

(db, data source)

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