L11n161

L11n160

L11n162

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n161 at Knotilus!

Link Presentations

[edit Notes on L11n161's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X18,11,19,12 X5,13,6,12 X6,21,1,22 X13,20,14,21 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+5t(2)t(1)-2t(1)+t(2)^{2}-3t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}+a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{3}+az$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-14a^{8}z^{6}+18a^{8}z^{4}-6a^{8}z^{2}-a^{8}+2a^{7}z^{9}-6a^{7}z^{7}-a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+7a^{6}z^{8}-30a^{6}z^{6}+34a^{6}z^{4}-9a^{6}z^{2}-3a^{6}+2a^{5}z^{9}-4a^{5}z^{7}-6a^{5}z^{5}+10a^{5}z^{3}-5a^{5}z+3a^{5}z^{-1}+4a^{4}z^{8}-15a^{4}z^{6}+16a^{4}z^{4}-4a^{4}z^{2}-3a^{4}+3a^{3}z^{7}-9a^{3}z^{5}+10a^{3}z^{3}-6a^{3}z+2a^{3}z^{-1}+a^{2}z^{6}+3az^{3}-2az+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   1 -1 0                 2   2 -2               3 2   -1 -4             4 1     3 -6           3 4       1 -8         4 3         1 -10       2 3           1 -12     2 4             -2 -14   1 2               1 -16   2                 -2 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n160 L11n162