L11a132

L11a131

L11a133

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a132 at Knotilus!

Link Presentations

[edit Notes on L11a132's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {4(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-4q^{9/2}+4q^{7/2}-4q^{5/2}+3q^{3/2}-{\frac {1}{q^{3/2}}}+q^{19/2}-2q^{17/2}+2q^{15/2}-3q^{13/2}+4q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-1}-z^{3}a^{-3}-z^{3}a^{-5}-z^{3}a^{-7}+az-za^{-1}-za^{-7}+za^{-9}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-10}-6z^{6}a^{-10}+10z^{4}a^{-10}-4z^{2}a^{-10}+2z^{9}a^{-9}-13z^{7}a^{-9}+26z^{5}a^{-9}-17z^{3}a^{-9}+2za^{-9}+z^{10}a^{-8}-5z^{8}a^{-8}+6z^{6}a^{-8}-z^{4}a^{-8}+3z^{9}a^{-7}-17z^{7}a^{-7}+29z^{5}a^{-7}-17z^{3}a^{-7}+2za^{-7}+z^{10}a^{-6}-5z^{8}a^{-6}+9z^{6}a^{-6}-10z^{4}a^{-6}+4z^{2}a^{-6}+z^{9}a^{-5}-3z^{7}a^{-5}+z^{5}a^{-5}+z^{8}a^{-4}-2z^{6}a^{-4}+z^{7}a^{-3}-z^{5}a^{-3}+z^{6}a^{-2}+z^{5}a^{-1}+az^{3}+z^{3}a^{-1}-2az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{4}-1$ (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 20                       1 -1 18                     1   1 16                   1 1   0 14                 2 1     1 12               2 1       -1 10             2 2         0 8           2 2           0 6         2 2             0 4       1 2               1 2     2 2                 0 0   1 3                   2 -2                         0 -4 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a131 L11a133