L11n61

L11n60

L11n62

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n61 at Knotilus!

Link Presentations

[edit Notes on L11n61's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X19,5,20,22 X15,21,16,20 X21,17,22,16 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -8, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+t(1)t(2)^{4}-3t(2)^{4}-2t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}-2t(2)^{2}-3t(1)t(2)+t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-2q^{5/2}+4q^{3/2}-5{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-8a^{3}z-4a^{3}z^{-1}+2az^{5}+8az^{3}-2z^{3}a^{-1}+9az+3az^{-1}-4za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+7a^{4}z^{4}+a^{4}z^{2}-2a^{4}+a^{3}z^{9}+a^{3}z^{7}-19a^{3}z^{5}+29a^{3}z^{3}-16a^{3}z+za^{-3}+4a^{3}z^{-1}+5a^{2}z^{8}-19a^{2}z^{6}+15a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}-3a^{2}-a^{-2}+az^{9}+2az^{7}+2z^{7}a^{-1}-21az^{5}-7z^{5}a^{-1}+28az^{3}+8z^{3}a^{-1}-13az-3za^{-1}+3az^{-1}+a^{-1}z^{-1}+3z^{8}-11z^{6}+9z^{4}+z^{2}-3$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 6                 2 2 4               2   -2 2             3 2   1 0           4 3     -1 -2         2 3 1     0 -4       3 4         1 -6     2 2           0 -8   1 4             3 -10   1               -1 -12 1                 1

Integral Khovanov Homology (db, data source)

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

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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