L11n56

L11n55

L11n57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n56 at Knotilus!

Link Presentations

[edit Notes on L11n56's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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