L11a146

L11a145

L11a147

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a146 at Knotilus!

Link Presentations

[edit Notes on L11a146's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X20,14,21,13 X22,16,7,15 X6,19,1,20 X18,11,19,12 X12,6,13,5 X14,22,15,21 X4,18,5,17
Gauss code	{1, -2, 3, -11, 9, -7}, {4, -1, 2, -3, 8, -9, 5, -10, 6, -4, 11, -8, 7, -5, 10, -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} , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-2t(1)^{2}t(2)^{5}+3t(1)t(2)^{5}-t(2)^{5}+2t(1)^{2}t(2)^{4}-5t(1)t(2)^{4}+2t(2)^{4}-2t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-2t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+9q^{7/2}-12q^{5/2}+13q^{3/2}-14{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-5z^{5}a^{-3}-7z^{3}a^{-3}-za^{-3}+a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-5az^{5}+17z^{5}a^{-1}-7az^{3}+15z^{3}a^{-1}-az+za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-7z^{4}a^{-6}+3z^{2}a^{-6}+4z^{7}a^{-5}-8z^{5}a^{-5}+3z^{3}a^{-5}+4z^{8}a^{-4}+a^{4}z^{6}-7z^{6}a^{-4}-3a^{4}z^{4}+2z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}-12z^{7}a^{-3}-10a^{3}z^{5}+20z^{5}a^{-3}+7a^{3}z^{3}-13z^{3}a^{-3}-a^{3}z+2za^{-3}+a^{-3}z^{-1}+2z^{10}a^{-2}+4a^{2}z^{8}-3z^{8}a^{-2}-12a^{2}z^{6}+7a^{2}z^{4}+7z^{4}a^{-2}-3a^{-2}+4az^{9}+8z^{9}a^{-1}-13az^{7}-32z^{7}a^{-1}+14az^{5}+53z^{5}a^{-1}-6az^{3}-31z^{3}a^{-1}-2az+za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{10}-3z^{8}-3z^{6}+8z^{4}-z^{2}-3$ (db)

Khovanov Homology

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

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     2   -2 10                   3 1   2 8                 6 2     -4 6               6 3       3 4             7 6         -1 2           7 6           1 0         5 8             3 -2       4 6               -2 -4     2 6                 4 -6   1 3                   -2 -8   2                     2 -10 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} $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=-3} ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ 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=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 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{7}$ 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=1} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ 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=2} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ 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}} ${\mathbb {Z} }^{6}$ $r=4$ 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}^{2}\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=5} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a145 L11a147