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

Visit L11a365 at Knotilus!

Link Presentations

[edit Notes on L11a365's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X6,11,7,12 X18,8,19,7 X22,18,11,17 X20,10,21,9 X8,20,9,19 X10,22,1,21 X4,15,5,16
Gauss code {1, -2, 3, -11, 4, -5, 6, -9, 8, -10}, {5, -1, 2, -3, 11, -4, 7, -6, 9, -8, 10, -7}
A Braid Representative
A Morse Link Presentation L11a365 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(t(2)^2 t(1)^2-t(2) t(1)^2+t(2) t(1)-t(1)+1\right) \left(t(1)^2 t(2)^2-t(1) t(2)^2+t(1) t(2)-t(2)+1\right)}{t(1)^2 t(2)^2} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+3 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z+a^3 z^{-1} -a z^9-8 a z^7+z^7 a^{-1} -23 a z^5+6 z^5 a^{-1} -28 a z^3+11 z^3 a^{-1} -13 a z+6 z a^{-1} -a z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-6 a^6 z^4+3 a^6 z^2+2 a^5 z^7-5 a^5 z^5+2 a^5 z^3+2 a^4 z^8-6 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-4 z^4 a^{-4} -a^4 z^2+3 z^2 a^{-4} +2 a^3 z^9-9 a^3 z^7+2 z^7 a^{-3} +18 a^3 z^5-8 z^5 a^{-3} -14 a^3 z^3+7 z^3 a^{-3} +6 a^3 z-z a^{-3} -a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+2 z^8 a^{-2} +a^2 z^6-7 z^6 a^{-2} +9 a^2 z^4+5 z^4 a^{-2} -6 a^2 z^2-z^2 a^{-2} +a^2+4 a z^9+2 z^9 a^{-1} -22 a z^7-9 z^7 a^{-1} +48 a z^5+16 z^5 a^{-1} -42 a z^3-16 z^3 a^{-1} +14 a z+6 z a^{-1} -a z^{-1} +z^{10}-3 z^8+z^6+6 z^4-6 z^2 (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 \
10           1-1
8          1 1
6         21 -1
4        31  2
2       32   -1
0      53    2
-2     34     1
-4    44      0
-6   24       2
-8  13        -2
-10 12         1
-12 1          -1
-141           1
Integral Khovanov Homology

(db, data source)

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