L11n138

L11n137

L11n139

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n138 at Knotilus!

Link Presentations

[edit Notes on L11n138's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+2a^{7}z-a^{5}z^{5}-4a^{5}z^{3}-4a^{5}z+a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-6z^{3}a^{9}+za^{9}-2z^{8}a^{8}+11z^{6}a^{8}-17z^{4}a^{8}+8z^{2}a^{8}-z^{9}a^{7}+4z^{7}a^{7}-2z^{5}a^{7}-z^{3}a^{7}-za^{7}-3z^{8}a^{6}+16z^{6}a^{6}-24z^{4}a^{6}+12z^{2}a^{6}-z^{9}a^{5}+5z^{7}a^{5}-8z^{5}a^{5}+9z^{3}a^{5}-5za^{5}-z^{8}a^{4}+5z^{6}a^{4}-7z^{4}a^{4}+4z^{2}a^{4}-z^{5}a^{3}+4z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-a^{2}-za+az^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ 0                 1 1 -2               1 1 0 -4             2     2 -6           1 2     1 -8         2 1       1 -10       1 1         0 -12     1 2           -1 -14   1 1             0 -16   1               -1 -18 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L11n137 L11n139