L11a301

L11a300

L11a302

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a301 at Knotilus!

Link Presentations

[edit Notes on L11a301's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,6,17,5 X20,14,21,13 X22,16,9,15 X14,22,15,21 X18,8,19,7 X6,18,7,17 X8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -6, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}\left(t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$9q^{9/2}-10q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-2q^{15/2}+4q^{13/2}-7q^{11/2}+6{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-8z^{7}a^{-3}+z^{7}a^{-5}+6z^{5}a^{-1}-24z^{5}a^{-3}+6z^{5}a^{-5}+12z^{3}a^{-1}-32z^{3}a^{-3}+12z^{3}a^{-5}+9za^{-1}-17za^{-3}+8za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-5z^{9}a^{-3}-3z^{9}a^{-5}+z^{8}a^{-2}-z^{8}a^{-4}-4z^{8}a^{-6}-2z^{8}-az^{7}+8z^{7}a^{-1}+23z^{7}a^{-3}+10z^{7}a^{-5}-4z^{7}a^{-7}+6z^{6}a^{-2}+12z^{6}a^{-4}+12z^{6}a^{-6}-3z^{6}a^{-8}+9z^{6}+5az^{5}-11z^{5}a^{-1}-46z^{5}a^{-3}-18z^{5}a^{-5}+10z^{5}a^{-7}-2z^{5}a^{-9}-10z^{4}a^{-2}-22z^{4}a^{-4}-17z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}-11z^{4}-7az^{3}+12z^{3}a^{-1}+51z^{3}a^{-3}+17z^{3}a^{-5}-12z^{3}a^{-7}+3z^{3}a^{-9}+9z^{2}a^{-2}+17z^{2}a^{-4}+6z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}+3z^{2}+2az-9za^{-1}-21za^{-3}-8za^{-5}+2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     1   1 14                   3 1   -2 12                 4 1     3 10               5 3       -2 8             5 4         1 6           4 5           1 4         5 5             0 2       3 6               3 0     1 3                 -2 -2   1 3                   2 -4   1                     -1 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a300 L11a302