L11a195

L11a194

L11a196

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a195 at Knotilus!

Link Presentations

[edit Notes on L11a195's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,15,21,16 X14,19,15,20 X4,18,5,17 X16,6,17,5 X12,21,13,22
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -8, 7, -10, 9, -5, 8, -7, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-4u^{2}v+2u^{2}-3uv^{4}+8uv^{3}-9uv^{2}+8uv-3u+2v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-12q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-4a^{3}z+8az-6za^{-1}+2za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+7a^{5}z^{7}-11a^{5}z^{5}+5a^{5}z^{3}-a^{5}z+7a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}-2a^{4}z^{4}-3z^{4}a^{-4}+3a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+4a^{3}z^{9}+5a^{3}z^{7}+3z^{7}a^{-3}-19a^{3}z^{5}-9z^{5}a^{-3}+14a^{3}z^{3}+9z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+a^{2}z^{10}+12a^{2}z^{8}+4z^{8}a^{-2}-24a^{2}z^{6}-8z^{6}a^{-2}+12a^{2}z^{4}+2z^{4}a^{-2}+3a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+7az^{9}+3z^{9}a^{-1}-4az^{7}+z^{7}a^{-1}-15az^{5}-17z^{5}a^{-1}+21az^{3}+22z^{3}a^{-1}-11az-12za^{-1}+3az^{-1}+2a^{-1}z^{-1}+z^{10}+9z^{8}-22z^{6}+13z^{4}+2z^{2}-3$ (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 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 8 2     6 2               8 5       -3 0             11 7         4 -2           9 9           0 -4         8 10             -2 -6       5 9               4 -8     3 8                 -5 -10   1 5                   4 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a194 L11a196