L11n202

L11n201

L11n203

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n202 at Knotilus!

Link Presentations

[edit Notes on L11n202's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,8,17,7 X11,18,12,19 X19,3,20,2 X3,12,4,13 X13,21,14,20 X5,15,6,14 X6,9,7,10 X15,22,16,9 X8,18,1,17 X21,4,22,5
Gauss code	{1, 4, -5, 11, -7, -8, 2, -10}, {8, -1, -3, 5, -6, 7, -9, -2, 10, 3, -4, 6, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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