L11n66

L11n65

L11n67

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n66 at Knotilus!

Link Presentations

[edit Notes on L11n66's Link Presentations]

Planar diagram presentation	X6172 X3,10,4,11 X7,14,8,15 X15,22,16,5 X9,17,10,16 X21,9,22,8 X17,21,18,20 X13,18,14,19 X19,12,20,13 X2536 X11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 6, -5, 2, -11, 9, -8, 3, -4, 5, -7, 8, -9, 7, -6, 4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{5}-5uv^{4}+8uv^{3}-4uv^{2}+uv+v^{4}-4v^{3}+8v^{2}-5v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {12}{q^{9/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{5/2}}}+q^{3/2}-{\frac {11}{q^{3/2}}}-{\frac {2}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-a^{7}z^{3}-4a^{7}z-3a^{7}z^{-1}+2a^{5}z^{5}+7a^{5}z^{3}+8a^{5}z+4a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-6a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}+az^{5}+2az^{3}$ (db)
Kauffman polynomial	$3a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+a^{8}z^{6}+4a^{8}z^{4}-4a^{8}z^{2}+a^{8}+5a^{7}z^{7}-13a^{7}z^{5}+25a^{7}z^{3}-16a^{7}z+3a^{7}z^{-1}+6a^{6}z^{8}-16a^{6}z^{6}+23a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+2a^{5}z^{9}+8a^{5}z^{7}-37a^{5}z^{5}+49a^{5}z^{3}-25a^{5}z+4a^{5}z^{-1}+11a^{4}z^{8}-29a^{4}z^{6}+23a^{4}z^{4}-9a^{4}z^{2}+2a^{4}+2a^{3}z^{9}+7a^{3}z^{7}-35a^{3}z^{5}+35a^{3}z^{3}-15a^{3}z+2a^{3}z^{-1}+5a^{2}z^{8}-11a^{2}z^{6}+2a^{2}z^{4}+a^{2}+4az^{7}-11az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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 4                   1 -1 2                 3   3 0               4 1   -3 -2             7 3     4 -4           6 5       -1 -6         7 6         1 -8       5 6           1 -10     4 7             -3 -12   2 6               4 -14   3                 -3 -16 2                   2

Integral Khovanov Homology (db, data source)

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

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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