L11a356

L11a355

L11a357

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a356 at Knotilus!

Link Presentations

[edit Notes on L11a356's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,5,17,6 X14,3,15,4 X20,10,21,9 X18,8,19,7 X22,17,11,18 X8,20,9,19 X4,15,5,16 X6,22,7,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -8, 2, -9, 5, -7, 4, -11}, {10, -1, 11, -3, 8, -2, 6, -5, 7, -4, 9, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-3u^{3}v^{3}+4u^{3}v^{2}-2u^{3}v-u^{2}v^{4}+4u^{2}v^{3}-7u^{2}v^{2}+4u^{2}v-u^{2}-2uv^{3}+4uv^{2}-3uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {10}{q^{7/2}}}+3q^{5/2}-{\frac {13}{q^{5/2}}}-6q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+5a^{3}z^{5}+4az^{5}-z^{5}a^{-1}-4a^{5}z^{3}+9a^{3}z^{3}+4az^{3}-3z^{3}a^{-1}-4a^{5}z+6a^{3}z-az-2za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}+2a^{7}z^{7}-7a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+2a^{6}z^{8}-4a^{6}z^{6}+a^{6}z^{2}+2a^{5}z^{9}-5a^{5}z^{7}+9a^{5}z^{5}-12a^{5}z^{3}+3a^{5}z+a^{4}z^{10}-a^{4}z^{6}+2a^{4}z^{4}-4a^{4}z^{2}+5a^{3}z^{9}-16a^{3}z^{7}+30a^{3}z^{5}+z^{5}a^{-3}-27a^{3}z^{3}-2z^{3}a^{-3}+10a^{3}z-a^{3}z^{-1}+a^{2}z^{10}+3a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}+16a^{2}z^{4}-6z^{4}a^{-2}-6a^{2}z^{2}+z^{2}a^{-2}+a^{2}+3az^{9}-4az^{7}+5z^{7}a^{-1}+az^{5}-12z^{5}a^{-1}+3az^{3}+9z^{3}a^{-1}+2az-3za^{-1}-az^{-1}+5z^{8}-12z^{6}+12z^{4}-4z^{2}$ (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                     2   -2 4                   4 1   3 2                 5 2     -3 0               7 4       3 -2             7 6         -1 -4           6 6           0 -6         5 8             3 -8       3 5               -2 -10     1 5                 4 -12   1 3                   -2 -14   1                     1 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11a355 L11a357