L10n20

L10n19

L10n21

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n20 at Knotilus!

Link Presentations

[edit Notes on L10n20's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -8, 9, -4, 6, -7, 4, -9, 8, 10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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