L11n178

L11n177

L11n179

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n178 at Knotilus!

Link Presentations

[edit Notes on L11n178's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X13,21,14,20 X3,10,4,11 X5,14,6,15 X7,16,8,17 X15,22,16,7 X11,4,12,5 X19,13,20,12 X21,1,22,6 X2,18,3,17
Gauss code	{1, -11, -4, 8, -5, 10}, {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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