L11n88

L11n87

L11n89

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n88 at Knotilus!

Link Presentations

[edit Notes on L11n88's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)^{3}\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {8}{q^{7/2}}}+8q^{5/2}-{\frac {12}{q^{5/2}}}-13q^{3/2}+{\frac {15}{q^{3/2}}}+15{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}-3a^{3}z^{3}+7az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-5a^{3}z+7az-4za^{-1}+za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-2z^{4}a^{-4}-7a^{4}z^{2}+z^{2}a^{-4}+a^{4}+10a^{3}z^{7}+4z^{7}a^{-3}-21a^{3}z^{5}-10z^{5}a^{-3}+28a^{3}z^{3}+8z^{3}a^{-3}-16a^{3}z-3za^{-3}+3a^{3}z^{-1}+10a^{2}z^{8}+6z^{8}a^{-2}-22a^{2}z^{6}-13z^{6}a^{-2}+27a^{2}z^{4}+5z^{4}a^{-2}-17a^{2}z^{2}+3a^{2}+a^{-2}+3az^{9}+3z^{9}a^{-1}+13az^{7}+7z^{7}a^{-1}-48az^{5}-37z^{5}a^{-1}+49az^{3}+35z^{3}a^{-1}-22az-13za^{-1}+4az^{-1}+2a^{-1}z^{-1}+16z^{8}-39z^{6}+28z^{4}-11z^{2}+2$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 3   3 6               5 1   -4 4             8 3     5 2           7 5       -2 0         10 8         2 -2       7 9           2 -4     5 8             -3 -6   3 7               4 -8   5                 -5 -10 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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