L11n77

L11n76

L11n78

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n77 at Knotilus!

Link Presentations

[edit Notes on L11n77's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{5}-4uv^{4}+7uv^{3}-3uv^{2}-3v^{3}+7v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+z^{5}a^{5}+4z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-11za^{3}-4a^{3}z^{-1}+2z^{5}a+7z^{3}a+7za+3az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{2}+5a^{7}z^{3}-a^{7}z+3a^{6}z^{6}-2a^{6}z^{4}+a^{6}+8a^{5}z^{7}-26a^{5}z^{5}+31a^{5}z^{3}-14a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-22a^{4}z^{6}+22a^{4}z^{4}-12a^{4}z^{2}+2a^{4}+2a^{3}z^{9}+6a^{3}z^{7}-42a^{3}z^{5}+53a^{3}z^{3}-26a^{3}z+4a^{3}z^{-1}+10a^{2}z^{8}-37a^{2}z^{6}+38a^{2}z^{4}-16a^{2}z^{2}+3a^{2}+2az^{9}-az^{7}+z^{7}a^{-1}-20az^{5}-4z^{5}a^{-1}+33az^{3}+6z^{3}a^{-1}-17az-4za^{-1}+3az^{-1}+a^{-1}z^{-1}+3z^{8}-12z^{6}+14z^{4}-5z^{2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 2   -2 2               4 1   3 0             4 2     -2 -2           6 4       2 -4         5 5         0 -6       4 5           -1 -8     3 5             2 -10   2 4               -2 -12   4                 4 -14 1                   -1

Integral Khovanov Homology (db, data source)

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

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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