L11n103

L11n102

L11n104

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n103 at Knotilus!

Link Presentations

[edit Notes on L11n103's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, -5, 6, -8, 3, -4, 5, -9, 7, -6, 4}

A Braid Representative
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} , $v$, 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} , ...)	$-{\frac {3t(1)t(2)^{3}+t(2)^{3}-3t(1)t(2)^{2}-3t(2)+t(1)+3}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}+{\frac {1}{q^{19/2}}}-{\frac {3}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {4}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+2a^{11}z+a^{11}z^{-1}+a^{9}z^{3}+4a^{9}z+2a^{9}z^{-1}-2a^{7}z^{5}-8a^{7}z^{3}-7a^{7}z-2a^{7}z^{-1}-a^{5}z^{5}-3a^{5}z^{3}-a^{5}z$ (db)
Kauffman polynomial	$a^{14}z^{8}-7a^{14}z^{6}+15a^{14}z^{4}-13a^{14}z^{2}+4a^{14}+a^{13}z^{9}-7a^{13}z^{7}+13a^{13}z^{5}-7a^{13}z^{3}+a^{13}z-a^{13}z^{-1}+2a^{12}z^{8}-17a^{12}z^{6}+41a^{12}z^{4}-34a^{12}z^{2}+9a^{12}+a^{11}z^{9}-7a^{11}z^{7}+12a^{11}z^{5}-2a^{11}z^{3}-a^{11}z-a^{11}z^{-1}+2a^{10}z^{8}-13a^{10}z^{6}+27a^{10}z^{4}-17a^{10}z^{2}+4a^{10}+3a^{9}z^{7}-13a^{9}z^{5}+21a^{9}z^{3}-12a^{9}z+2a^{9}z^{-1}+a^{8}z^{8}-a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+3a^{7}z^{7}-11a^{7}z^{5}+13a^{7}z^{3}-9a^{7}z+2a^{7}z^{-1}+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-3a^{5}z^{3}+a^{5}z$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     2 1 -1 -8                   2     2 -10                 2 2     0 -12             1 4 2       1 -14             1 3         2 -16         1 4 3           0 -18       1   1             2 -20       1 3               -2 -22   1 1                   0 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }_{2}$ 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}} $r=-9$ 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}} $r=-8$ 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} ${\mathbb {Z} }$ 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=-7} ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ 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}\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}} $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ 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 r=-3} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ 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}^{2}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11n102 L11n104