L11n170

L11n169

L11n171

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n170 at Knotilus!

Link Presentations

[edit Notes on L11n170's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,21,14,20 X19,13,20,12 X22,16,7,15 X4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 9, -8, -5, 10, -7, 4, 2, -9, 8, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-u^{2}v+3uv^{3}-5uv^{2}+3uv-v^{3}+3v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$-7q^{9/2}+9q^{7/2}-10q^{5/2}+7q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-7{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+3z^{3}a^{-5}+3za^{-5}+2a^{-5}z^{-1}-z^{7}a^{-3}-5z^{5}a^{-3}-9z^{3}a^{-3}-8za^{-3}-5a^{-3}z^{-1}+z^{5}a^{-1}+3z^{3}a^{-1}+4za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-z^{8}a^{-2}-4z^{8}a^{-4}-3z^{8}a^{-6}-3z^{7}a^{-5}-3z^{7}a^{-7}-2z^{6}a^{-2}+5z^{6}a^{-4}+6z^{6}a^{-6}-z^{6}a^{-8}-7z^{5}a^{-1}-5z^{5}a^{-3}+11z^{5}a^{-5}+9z^{5}a^{-7}+3z^{4}a^{-4}+3z^{4}a^{-6}+3z^{4}a^{-8}-3z^{4}-az^{3}+9z^{3}a^{-1}+14z^{3}a^{-3}-2z^{3}a^{-5}-6z^{3}a^{-7}+5z^{2}a^{-2}+2z^{2}a^{-4}-6z^{2}a^{-6}-3z^{2}a^{-8}-7za^{-1}-10za^{-3}-3za^{-5}-5a^{-2}-5a^{-4}+a^{-8}+3a^{-1}z^{-1}+5a^{-3}z^{-1}+2a^{-5}z^{-1}$ (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 $j$, alternation over $r$).

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

Integral Khovanov Homology (db, data source)

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

L11n169 L11n171