L11n223

L11n222

L11n224

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n223 at Knotilus!

Link Presentations

[edit Notes on L11n223's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,8,19,7 X16,9,17,10 X22,15,9,16 X21,5,22,4 X5,14,6,15 X13,20,14,21 X8,18,1,17 X6,20,7,19
Gauss code	{1, -2, 3, 7, -8, -11, 4, -10}, {5, -1, 2, -3, -9, 8, 6, -5, 10, -4, 11, 9, -7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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} , ...)	${\frac {(u-1)(v-1)\left(u^{2}v+2uv^{2}+2u+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {8}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+z^{5}a^{5}+2z^{3}a^{5}-a^{5}z^{-1}+2z^{5}a^{3}+7z^{3}a^{3}+7za^{3}+3a^{3}z^{-1}-2z^{3}a-5za-2az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-3a^{9}z^{3}+3a^{8}z^{6}-5a^{8}z^{4}+3a^{8}z^{2}+4a^{7}z^{7}-11a^{7}z^{5}+15a^{7}z^{3}-4a^{7}z+3a^{6}z^{8}-8a^{6}z^{6}+12a^{6}z^{4}-5a^{6}z^{2}+a^{6}+a^{5}z^{9}+a^{5}z^{7}-6a^{5}z^{5}+7a^{5}z^{3}-a^{5}z^{-1}+4a^{4}z^{8}-13a^{4}z^{6}+20a^{4}z^{4}-16a^{4}z^{2}+3a^{4}+a^{3}z^{9}-3a^{3}z^{7}+10a^{3}z^{5}-21a^{3}z^{3}+13a^{3}z-3a^{3}z^{-1}+a^{2}z^{8}-2a^{2}z^{6}+2a^{2}z^{4}-6a^{2}z^{2}+3a^{2}+3az^{5}-10az^{3}+9az-2az^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums $\chi$ (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 2                   2 2 0                 1   -1 -2               5 2   3 -4             4 3     -1 -6           4 3       1 -8         4 4         0 -10       3 4           -1 -12     1 4             3 -14   1 3               -2 -16   1                 1 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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