L10n10

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n10 at Knotilus!

Link Presentations

[edit Notes on L10n10's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X3849 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X13,2,14,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -6, 9, -8, 4, -10, -2, 3, 6, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-5a^{7}z^{3}-6a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+11a^{5}z^{3}+9a^{5}z+4a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-za^{11}-z^{2}a^{10}+a^{10}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-z^{6}a^{8}+4z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-2z^{7}a^{7}+11z^{5}a^{7}-18z^{3}a^{7}+12za^{7}-3a^{7}z^{-1}-z^{8}a^{6}+4z^{6}a^{6}-z^{4}a^{6}-5z^{2}a^{6}+3a^{6}-3z^{7}a^{5}+17z^{5}a^{5}-28z^{3}a^{5}+17za^{5}-4a^{5}z^{-1}-z^{8}a^{4}+5z^{6}a^{4}-5z^{4}a^{4}-2z^{2}a^{4}+2a^{4}-z^{7}a^{3}+6z^{5}a^{3}-11z^{3}a^{3}+8za^{3}-2a^{3}z^{-1}$ (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             2 1   1 -6         1 1 1     1 -8         2 1       1 -10     1 2 1         0 -12   1 2 2           -1 -14   1 1             0 -16 1 1               0 -18 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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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