L11n181

L11n180

L11n182

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n181 at Knotilus!

Link Presentations

[edit Notes on L11n181's Link Presentations]

Planar diagram presentation	X8192 X3,10,4,11 X5,14,6,15 X16,8,17,7 X22,18,7,17 X15,13,16,12 X20,10,21,9 X11,19,12,18 X13,6,14,1 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, 7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, -7, 11, -5}

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} , ...)	$-{\frac {-u^{2}v^{2}+u^{2}v+u^{2}-uv^{3}+3uv^{2}-uv+v^{4}+v^{3}-v^{2}}{uv^{2}}}$ (db)
Jones 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 -q^{7/2}+2 q^{5/2}-\frac{2}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{11/2}}+3 \sqrt{q}-\frac{3}{\sqrt{q}}} (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}-za^{-3}+az^{3}+z^{3}a^{-1}+az+az^{-1}+za^{-1}$ (db)
Kauffman 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^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+2 a^5 z^{-1} -2 a^4 z^4+6 a^4 z^2-3 a^4-2 a^3 z^5+z^5 a^{-3} +9 a^3 z^3-3 z^3 a^{-3} -9 a^3 z+z a^{-3} +3 a^3 z^{-1} +2 z^6 a^{-2} -7 z^4 a^{-2} +4 a^2 z^2+5 z^2 a^{-2} -3 a^2+z^7 a^{-1} +2 a z^5-2 z^5 a^{-1} -3 a z^3-z^3 a^{-1} +z a^{-1} +a z^{-1} +2 z^6-5 z^4+3 z^2-1} (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 $j$, 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   1   -1 4                 1 1   0 2               2 1     -1 0           1 2 1       0 -2           2 3         1 -4       1 2 1           0 -6         2             2 -8     1 1               0 -10 1                     1 -12 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L11n180 L11n182