L10n26

L10n25

L10n27

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n26 at Knotilus!

Link Presentations

[edit Notes on L10n26's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,17,5,18 X18,14,19,13 X12,20,13,19 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, 3, -8, 7, 4, -5, -3, 6, -7, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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