L11a194

L11a193

L11a195

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a194 at Knotilus!

Link Presentations

[edit Notes on L11a194's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3u^{2}v^{2}-5u^{2}v+2u^{2}-5uv^{2}+9uv-5u+2v^{2}-5v+3}{uv}}$ (db)
Jones polynomial	$q^{21/2}-3q^{19/2}+5q^{17/2}-8q^{15/2}+11q^{13/2}-12q^{11/2}+12q^{9/2}-11q^{7/2}+7q^{5/2}-5q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}-z^{5}a^{-7}-2z^{3}a^{-7}-2za^{-7}-z^{5}a^{-5}+2za^{-5}-z^{5}a^{-3}-2z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-5z^{9}a^{-7}-3z^{9}a^{-9}-2z^{8}a^{-4}-z^{8}a^{-6}-3z^{8}a^{-8}-4z^{8}a^{-10}-2z^{7}a^{-3}+3z^{7}a^{-5}+15z^{7}a^{-7}+7z^{7}a^{-9}-3z^{7}a^{-11}-2z^{6}a^{-2}+5z^{6}a^{-6}+17z^{6}a^{-8}+13z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}-7z^{5}a^{-5}-19z^{5}a^{-7}-z^{5}a^{-9}+10z^{5}a^{-11}+4z^{4}a^{-2}+3z^{4}a^{-4}-10z^{4}a^{-6}-23z^{4}a^{-8}-11z^{4}a^{-10}+3z^{4}a^{-12}+3z^{3}a^{-1}+7z^{3}a^{-3}+8z^{3}a^{-5}+7z^{3}a^{-7}-4z^{3}a^{-9}-7z^{3}a^{-11}-z^{2}a^{-2}+5z^{2}a^{-6}+9z^{2}a^{-8}+4z^{2}a^{-10}-z^{2}a^{-12}-3za^{-1}-5za^{-3}-3za^{-5}-za^{-7}+za^{-9}+za^{-11}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 22                       1 -1 20                     2   2 18                   3 1   -2 16                 5 2     3 14               6 3       -3 12             6 5         1 10           6 6           0 8         5 6             -1 6       3 7               4 4     2 4                 -2 2   1 4                   3 0   1                     -1 -2 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a193 L11a195