L11n97

L11n96

L11n98

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n97 at Knotilus!

Link Presentations

[edit Notes on L11n97's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X16,8,17,7 X18,10,19,9 X8,18,9,17 X19,22,20,5 X13,20,14,21 X21,14,22,15 X10,16,11,15 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 4, -9, 11, -2, -7, 8, 9, -3, 5, -4, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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