L11n8

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n8 at Knotilus!

Link Presentations

[edit Notes on L11n8's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, 11, -2, -3, 9, -8, 7, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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 v} , 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 w} , ...)	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 -\frac{(u-1) (v-1)^3}{\sqrt{u} v^{3/2}}} (db)
Jones polynomial	$-q^{3/2}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	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 a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +2 a z^3+4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} } (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-2a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-2a^{7}z^{7}+a^{5}z^{7}+2a^{3}z^{7}-az^{7}-a^{8}z^{6}+7a^{6}z^{6}+18a^{4}z^{6}+10a^{2}z^{6}+8a^{7}z^{5}+10a^{5}z^{5}+6a^{3}z^{5}+4az^{5}+4a^{8}z^{4}-3a^{6}z^{4}-24a^{4}z^{4}-19a^{2}z^{4}-2z^{4}-7a^{7}z^{3}-19a^{5}z^{3}-21a^{3}z^{3}-10az^{3}-z^{3}a^{-1}-3a^{8}z^{2}-3a^{6}z^{2}+8a^{4}z^{2}+11a^{2}z^{2}+3z^{2}+3a^{7}z+12a^{5}z+16a^{3}z+9az+2za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

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

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   1 1 2                 1   -1 0               3 1   2 -2             3 3     0 -4           3 2 1     2 -6         3 4         1 -8       2 3           -1 -10     1 3 1           1 -12   1 2               -1 -14   1                 1 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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