L10n13

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n13 at Knotilus!

Link Presentations

[edit Notes on L10n13's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X17,2,18,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 9, -8, 6, -7, 4, -9, 8, -10, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}+t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-5a^{7}z^{3}-7a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+12a^{5}z^{3}+11a^{5}z+4a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-7a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z+2a^{10}z^{2}-a^{10}+a^{9}z^{5}-a^{9}z^{3}-a^{9}z+a^{9}z^{-1}+3a^{8}z^{6}-11a^{8}z^{4}+10a^{8}z^{2}-3a^{8}+3a^{7}z^{7}-13a^{7}z^{5}+17a^{7}z^{3}-11a^{7}z+3a^{7}z^{-1}+a^{6}z^{8}-a^{6}z^{6}-9a^{6}z^{4}+12a^{6}z^{2}-3a^{6}+4a^{5}z^{7}-20a^{5}z^{5}+30a^{5}z^{3}-18a^{5}z+4a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+2a^{4}z^{4}+4a^{4}z^{2}-2a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 0                 1 1 -2                   0 -4             3 1   2 -6           1 1     0 -8         2 2       0 -10     1 2 1         0 -12     2 2           0 -14   1 2             1 -16   1               -1 -18 1                 1

Integral Khovanov Homology (db, data source)

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

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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