L11a443

L11a442

L11a444

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a443 at Knotilus!

Link Presentations

[edit Notes on L11a443's Link Presentations]

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

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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