L11n52

L11n51

L11n53

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n52 at Knotilus!

Link Presentations

[edit Notes on L11n52's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X16,9,17,10 X8,17,9,18 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, -4, 9, -8, -5, 6, 4, -7, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+3z^{3}a+3za+az^{-1}-z^{3}a^{-1}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+8a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-5a^{4}z^{2}+a^{4}+a^{3}z^{9}-16a^{3}z^{5}+27a^{3}z^{3}-15a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+5a^{2}z^{8}-22a^{2}z^{6}+26a^{2}z^{4}+z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+3a^{2}+a^{-2}+az^{9}+az^{7}+2z^{7}a^{-1}-19az^{5}-8z^{5}a^{-1}+28az^{3}+9z^{3}a^{-1}-12az-za^{-1}+az^{-1}-a^{-1}z^{-1}+3z^{8}-13z^{6}+16z^{4}-8z^{2}+2$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 6                 2 2 4               1   -1 2             4 2   2 0           3 3     0 -2         2 3 1     0 -4       3 3         0 -6     1 2           -1 -8   1 3             2 -10   1               -1 -12 1                 1

Integral Khovanov Homology (db, data source)

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

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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