L11n218

L11n217

L11n219

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n218 at Knotilus!

Link Presentations

[edit Notes on L11n218's Link Presentations]

Planar diagram presentation	X10,1,11,2 X8,9,1,10 X3,12,4,13 X22,16,9,15 X17,3,18,2 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X19,7,20,6 X7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, 9, 3, -8, 7, 4, -9, -5, 11, -10, 8, -6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(u-1) (u+1)^2 (v-1)}{u^{3/2} \sqrt{v}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-q^{5/2}-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -5 z^3 a^{-3} -6 z a^{-3} -3 a^{-3} z^{-1} +a z+2 z a^{-1} +2 a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -2 z^8 a^{-4} -z^8 a^{-6} +7 z^7 a^{-3} +7 z^7 a^{-5} +7 z^6 a^{-2} +14 z^6 a^{-4} +7 z^6 a^{-6} -15 z^5 a^{-3} -15 z^5 a^{-5} -14 z^4 a^{-2} -28 z^4 a^{-4} -15 z^4 a^{-6} -z^4-a z^3+z^3 a^{-1} +15 z^3 a^{-3} +13 z^3 a^{-5} +10 z^2 a^{-2} +18 z^2 a^{-4} +11 z^2 a^{-6} +3 z^2+2 a z-3 z a^{-1} -11 z a^{-3} -6 z a^{-5} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 14                   1 -1 12                     0 10               1 1   0 8           1 1       0 6         1   1       2 4         2 1         1 2     2 1             1 0   1 2 1             0 -2                     0 -4 1                   1

Integral Khovanov Homology (db, data source)

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

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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