L11n206

L11n205

L11n207

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n206 at Knotilus!

Link Presentations

[edit Notes on L11n206's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,8,17,7 X18,12,19,11 X19,3,20,2 X3,12,4,13 X13,21,14,20 X14,5,15,6 X6,9,7,10 X22,16,9,15 X8,18,1,17 X4,22,5,21
Gauss code	{1, 4, -5, -11, 7, -8, 2, -10}, {8, -1, 3, 5, -6, -7, 9, -2, 10, -3, -4, 6, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(2)-1) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-4 q^{9/2}+5 q^{7/2}-6 q^{5/2}+4 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+3 q^{11/2}-4 \sqrt{q}+\frac{2}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^5 a^{-5} +4 z^3 a^{-5} +4 z a^{-5} + a^{-5} z^{-1} -z^7 a^{-3} -6 z^5 a^{-3} -12 z^3 a^{-3} -9 z a^{-3} -3 a^{-3} z^{-1} +z^5 a^{-1} +4 z^3 a^{-1} +5 z a^{-1} +2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$z^6 a^{-8} -4 z^4 a^{-8} +3 z^2 a^{-8} +2 z^7 a^{-7} -8 z^5 a^{-7} +7 z^3 a^{-7} -z a^{-7} +2 z^8 a^{-6} -8 z^6 a^{-6} +9 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} +z^9 a^{-5} -3 z^7 a^{-5} +3 z^5 a^{-5} -5 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +3 z^8 a^{-4} -13 z^6 a^{-4} +21 z^4 a^{-4} -14 z^2 a^{-4} +3 a^{-4} +z^9 a^{-3} -5 z^7 a^{-3} +14 z^5 a^{-3} -18 z^3 a^{-3} +11 z a^{-3} -3 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +10 z^4 a^{-2} -8 z^2 a^{-2} +3 a^{-2} +3 z^5 a^{-1} +a z^3-5 z^3 a^{-1} -a z+5 z a^{-1} -2 a^{-1} z^{-1} +2 z^4-2 z^2$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 1   1 12               2 1   -1 10             2 1     1 8           3 2       -1 6         3 2         1 4       1 3           2 2     3 3             0 0   1 3               2 -2   1                 -1 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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). Back to the top.

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