L11a69

L11a68

L11a70

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a69 at Knotilus!

Link Presentations

[edit Notes on L11a69's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,10,15,9 X22,20,5,19 X20,11,21,12 X10,21,11,22 X18,15,19,16 X16,7,17,8 X8,17,9,18 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -9, 3, -6, 5, -2, 11, -3, 7, -8, 9, -7, 4, -5, 6, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+8az^{3}-2z^{3}a^{-1}+6az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-9a^{3}z^{9}-5az^{9}-5a^{6}z^{8}-16a^{4}z^{8}-20a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-2a^{5}z^{7}-9az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+41a^{4}z^{6}+47a^{2}z^{6}-4z^{6}a^{-2}+12z^{6}+7a^{7}z^{5}+21a^{5}z^{5}+36a^{3}z^{5}+36az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-3a^{6}z^{4}-32a^{4}z^{4}-37a^{2}z^{4}+5z^{4}a^{-2}-6z^{4}-6a^{7}z^{3}-22a^{5}z^{3}-41a^{3}z^{3}-35az^{3}-9z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-2a^{6}z^{2}+9a^{4}z^{2}+11a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+3a^{7}z+12a^{5}z+18a^{3}z+13az+4za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     3   -3 4                   6 1   5 2                 8 3     -5 0               12 6       6 -2             12 10         -2 -4           10 10           0 -6         8 12             4 -8       6 10               -4 -10     2 8                 6 -12   1 6                   -5 -14   2                     2 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a68 L11a70