L11n249

L11n248

L11n250

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n249 at Knotilus!

Link Presentations

[edit Notes on L11n249's Link Presentations]

Planar diagram presentation	X12,1,13,2 X9,19,10,18 X5,14,6,15 X11,6,12,7 X15,11,16,22 X7,20,8,21 X3948 X21,17,22,16 X17,4,18,5 X10,13,1,14 X19,3,20,2
Gauss code	{1, 11, -7, 9, -3, 4, -6, 7, -2, -10}, {-4, -1, 10, 3, -5, 8, -9, 2, -11, 6, -8, 5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, $w$, ...)	${\frac {u^{3}v^{3}-3u^{3}v^{2}+2u^{3}v-2u^{2}v^{3}+7u^{2}v^{2}-7u^{2}v+u^{2}+uv^{3}-7uv^{2}+7uv-2u+2v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {7}{q^{7/2}}}+8q^{5/2}-{\frac {12}{q^{5/2}}}-12q^{3/2}+{\frac {14}{q^{3/2}}}+15{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+6az^{3}-5z^{3}a^{-1}+3az-3za^{-1}$ (db)
Kauffman polynomial	$-3az^{9}-3z^{9}a^{-1}-8a^{2}z^{8}-6z^{8}a^{-2}-14z^{8}-8a^{3}z^{7}-8az^{7}-4z^{7}a^{-1}-4z^{7}a^{-3}-3a^{4}z^{6}+13a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+31z^{6}+13a^{3}z^{5}+27az^{5}+24z^{5}a^{-1}+10z^{5}a^{-3}-3a^{4}z^{4}-10a^{2}z^{4}-7z^{4}a^{-2}+2z^{4}a^{-4}-16z^{4}-6a^{5}z^{3}-15a^{3}z^{3}-18az^{3}-16z^{3}a^{-1}-7z^{3}a^{-3}+a^{4}z^{2}+3a^{2}z^{2}-z^{2}a^{-4}+3z^{2}+5a^{5}z+7a^{3}z+4az+3za^{-1}+za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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 $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               5 1   -4 4             7 3     4 2           8 5       -3 0         8 7         1 -2       7 9           2 -4     5 7             -2 -6   2 7               5 -8 1 5                 -4 -10 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ 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}^{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}} $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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=-2} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ 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}^{5}} $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ 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 r=0} ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ 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}^{7}\oplus{\mathbb Z}_2^{8}} ${\mathbb {Z} }^{8}$ $r=2$ 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}^{5}\oplus{\mathbb Z}_2^{7}} ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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