L11n133

L11n132

L11n134

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n133 at Knotilus!

Link Presentations

[edit Notes on L11n133's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X9,16,10,17 X13,20,14,21 X15,22,16,7 X19,4,20,5 X21,14,22,15
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, 4, -2, -10, 8, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{6}+uv^{4}-uv^{3}+uv^{2}+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{19/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-3za^{13}-2a^{13}z^{-1}+z^{7}a^{11}+8z^{5}a^{11}+20z^{3}a^{11}+18za^{11}+5a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-18za^{9}-3a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{16}z^{2}-a^{16}+a^{15}z^{3}-a^{15}z+a^{14}z^{2}-a^{13}z^{3}+4a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+20a^{12}z^{4}-18a^{12}z^{2}+5a^{12}+a^{11}z^{9}-9a^{11}z^{7}+28a^{11}z^{5}-38a^{11}z^{3}+23a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+20a^{10}z^{4}-18a^{10}z^{2}+5a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+18a^{9}z-3a^{9}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                   1 1 -10                   1 1 -12               1     1 -14           1         1 -16           1 1       0 -18       2 1           1 -20       1 1           0 -22   1 1               0 -24                     0 -26 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n132 L11n134