L11a9

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(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a9 at Knotilus!

Link Presentations

[edit Notes on L11a9's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,5,13 X14,22,15,21 X18,12,19,11 X20,10,21,9 X10,20,11,19 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 9, -10, 8, -4, 6, -7, 11, -2, 3, -8, 10, -9, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(4v^{2}-7v+4\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-16q^{9/2}+18q^{7/2}-20q^{5/2}+{\frac {1}{q^{5/2}}}+17q^{3/2}-{\frac {4}{q^{3/2}}}-q^{17/2}+3q^{15/2}-6q^{13/2}+12q^{11/2}-14{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-7}-za^{-7}+z^{5}a^{-5}+z^{3}a^{-5}+2za^{-5}+a^{-5}z^{-1}+2z^{5}a^{-3}+2z^{3}a^{-3}-2za^{-3}-3a^{-3}z^{-1}+z^{5}a^{-1}-az^{3}+za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-4z^{5}a^{-9}+4z^{3}a^{-9}+3z^{8}a^{-8}-12z^{6}a^{-8}+16z^{4}a^{-8}-8z^{2}a^{-8}+4z^{9}a^{-7}-13z^{7}a^{-7}+13z^{5}a^{-7}-6z^{3}a^{-7}+2za^{-7}+2z^{10}a^{-6}+3z^{8}a^{-6}-26z^{6}a^{-6}+34z^{4}a^{-6}-17z^{2}a^{-6}+a^{-6}+11z^{9}a^{-5}-29z^{7}a^{-5}+20z^{5}a^{-5}-7z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+2z^{10}a^{-4}+12z^{8}a^{-4}-42z^{6}a^{-4}+36z^{4}a^{-4}-13z^{2}a^{-4}+3a^{-4}+7z^{9}a^{-3}-3z^{7}a^{-3}-18z^{5}a^{-3}+13z^{3}a^{-3}+3za^{-3}-3a^{-3}z^{-1}+12z^{8}a^{-2}-20z^{6}a^{-2}+a^{2}z^{4}+11z^{4}a^{-2}-4z^{2}a^{-2}+3a^{-2}+12z^{7}a^{-1}+4az^{5}-17z^{5}a^{-1}-2az^{3}+8z^{3}a^{-1}+za^{-1}-2a^{-1}z^{-1}+8z^{6}-6z^{4}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     2   -2 14                   4 1   3 12                 8 2     -6 10               8 4       4 8             10 8         -2 6           10 8           2 4         7 10             3 2       7 10               -3 0     3 9                 6 -2   1 5                   -4 -4   3                     3 -6 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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