L10n34

L10n33

L10n35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n34 at Knotilus!

Link Presentations

[edit Notes on L10n34's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,8,17,7 X17,20,18,5 X11,19,12,18 X19,11,20,10 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 7, 6, -5, -2, 10, -7, 8, -3, -4, 5, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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