L11n146

L11n145

L11n147

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n146 at Knotilus!

Link Presentations

[edit Notes on L11n146's Link Presentations]

Planar diagram presentation	X8192 X9,19,10,18 X6718 X19,7,20,22 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X13,21,14,20 X21,15,22,14 X17,2,18,3
Gauss code	{1, 11, -6, 7, -5, -3}, {3, -1, -2, 6, -8, 5, -9, 10, -7, 8, -11, 2, -4, 9, -10, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)+t(2)-1)(t(2)t(1)-t(1)-t(2))\left(t(2)^{2}-t(2)+1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+4q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {5}{q^{5/2}}}+8q^{3/2}-{\frac {8}{q^{3/2}}}-10{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{3}-z^{3}a^{-3}-a^{3}z+a^{-3}z^{-1}+az^{5}+z^{5}a^{-1}+2az^{3}+z^{3}a^{-1}+2az-2za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+a^{4}z^{6}-3a^{4}z^{4}+4z^{4}a^{-4}+2a^{4}z^{2}-2z^{2}a^{-4}-a^{-4}+3a^{3}z^{7}+z^{7}a^{-3}-10a^{3}z^{5}+2z^{5}a^{-3}+9a^{3}z^{3}-z^{3}a^{-3}-3a^{3}z+a^{-3}z^{-1}+3a^{2}z^{8}+2z^{8}a^{-2}-7a^{2}z^{6}-3z^{6}a^{-2}+5z^{4}a^{-2}+3a^{2}z^{2}-3a^{-2}+az^{9}+z^{9}a^{-1}+4az^{7}+2z^{7}a^{-1}-19az^{5}-7z^{5}a^{-1}+18az^{3}+7z^{3}a^{-1}-8az-5za^{-1}+2az^{-1}+3a^{-1}z^{-1}+5z^{8}-11z^{6}+4z^{4}+3z^{2}-3$ (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 10                   1 1 8                 3   -3 6               3 1   2 4             5 3     -2 2           5 3       2 0         4 6         2 -2       4 4           0 -4     2 5             3 -6   1 3               -2 -8   2                 2 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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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