L11n230

L11n229

L11n231

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n230 at Knotilus!

Link Presentations

[edit Notes on L11n230's Link Presentations]

Planar diagram presentation	X10,1,11,2 X11,17,12,16 X5,21,6,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X17,9,18,22 X21,19,22,18 X8,9,1,10 X19,5,20,4 X15,2,16,3
Gauss code	{1, 11, -4, 10, -3, 6, -5, -9}, {9, -1, -2, 4, -6, 5, -11, 2, -7, 8, -10, 3, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{2}-u^{3}v+2u^{2}v^{3}-4u^{2}v^{2}+5u^{2}v-2u^{2}-2uv^{3}+5uv^{2}-4uv+2u-v^{2}+v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$2q^{5/2}-5q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {10}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-2a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}-2az^{5}-6az^{3}+2z^{3}a^{-1}-5az+3za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-6a^{5}z^{5}+5a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-6a^{4}z^{6}+7a^{4}z^{4}-3a^{4}z^{2}-a^{4}+a^{3}z^{9}+4a^{3}z^{7}-13a^{3}z^{5}+16a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}+5a^{2}z^{8}-10a^{2}z^{6}+11a^{2}z^{4}+3z^{4}a^{-2}-4a^{2}z^{2}-4z^{2}a^{-2}+az^{9}+2az^{7}+z^{7}a^{-1}-3az^{5}+3z^{5}a^{-1}-8z^{3}a^{-1}+2az+3za^{-1}+2z^{8}-2z^{6}+3z^{4}-4z^{2}$ (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 6                   2 -2 4                 3   3 2               4 2   -2 0             6 3     3 -2           5 5       0 -4         5 5         0 -6       3 5           2 -8     2 5             -3 -10   1 4               3 -12   1                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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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