L11n159

L11n158

L11n160

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n159 at Knotilus!

Link Presentations

[edit Notes on L11n159's Link Presentations]

Planar diagram presentation	X8192 X7,17,8,16 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X21,1,22,6 X20,14,21,13 X17,7,18,22 X4,20,5,19
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 $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+u^{2}v^{2}-uv^{4}+uv^{2}-u+v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$2q^{15/2}-2q^{13/2}+2q^{11/2}-4q^{9/2}+2q^{7/2}-3q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}+z^{5}a^{-3}-5z^{5}a^{-5}+z^{5}a^{-7}+4z^{3}a^{-3}-6z^{3}a^{-5}+4z^{3}a^{-7}+3za^{-3}-2za^{-5}+za^{-7}-za^{-9}+a^{-3}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-5}-z^{9}a^{-7}-2z^{8}a^{-4}-3z^{8}a^{-6}-z^{8}a^{-8}-z^{7}a^{-3}+3z^{7}a^{-5}+4z^{7}a^{-7}+10z^{6}a^{-4}+15z^{6}a^{-6}+5z^{6}a^{-8}+5z^{5}a^{-3}+4z^{5}a^{-5}-z^{5}a^{-7}-12z^{4}a^{-4}-19z^{4}a^{-6}-7z^{4}a^{-8}-7z^{3}a^{-3}-12z^{3}a^{-5}-5z^{3}a^{-7}+2z^{2}a^{-4}+8z^{2}a^{-6}+7z^{2}a^{-8}+z^{2}a^{-10}+4za^{-3}+6za^{-5}+3za^{-7}+za^{-9}+a^{-4}-3a^{-6}-5a^{-8}-2a^{-10}-a^{-3}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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 χ 16               2 -2 14             1 1 0 12           2 2   0 10         2 1 1   2 8       1 3       2 6     2 2 1       1 4   1 2           1 2   1             -1 0 1               1

Integral Khovanov Homology (db, data source)

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

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.

L11n158 L11n160