L11a233

L11a232

L11a234

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a233 at Knotilus!

Link Presentations

[edit Notes on L11a233's Link Presentations]

Planar diagram presentation	X8192 X12,4,13,3 X18,6,19,5 X20,12,21,11 X22,16,7,15 X10,18,11,17 X16,22,17,21 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -6, 4, -2, 11, -8, 5, -7, 6, -3, 9, -4, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {3u^{2}v^{3}-7u^{2}v^{2}+6u^{2}v-2u^{2}+2uv^{4}-11uv^{3}+17uv^{2}-11uv+2u-2v^{4}+6v^{3}-7v^{2}+3v}{uv^{2}}}$ (db)
Jones polynomial	$25q^{9/2}-23q^{7/2}+16q^{5/2}-10q^{3/2}+q^{21/2}-4q^{19/2}+10q^{17/2}-16q^{15/2}+22q^{13/2}-26q^{11/2}+4{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}+z^{3}a^{-7}-2a^{-7}z^{-1}-3z^{5}a^{-5}-5z^{3}a^{-5}-3za^{-5}-z^{5}a^{-3}+2z^{3}a^{-3}+4za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-8z^{5}a^{-11}+4z^{3}a^{-11}+8z^{8}a^{-10}-18z^{6}a^{-10}+14z^{4}a^{-10}-7z^{2}a^{-10}+2a^{-10}+8z^{9}a^{-9}-13z^{7}a^{-9}+5z^{5}a^{-9}-4z^{3}a^{-9}+2za^{-9}-a^{-9}z^{-1}+3z^{10}a^{-8}+13z^{8}a^{-8}-43z^{6}a^{-8}+39z^{4}a^{-8}-19z^{2}a^{-8}+5a^{-8}+17z^{9}a^{-7}-31z^{7}a^{-7}+14z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}-2a^{-7}z^{-1}+3z^{10}a^{-6}+17z^{8}a^{-6}-48z^{6}a^{-6}+41z^{4}a^{-6}-14z^{2}a^{-6}+3a^{-6}+9z^{9}a^{-5}-5z^{7}a^{-5}-14z^{5}a^{-5}+18z^{3}a^{-5}-5za^{-5}+12z^{8}a^{-4}-20z^{6}a^{-4}+14z^{4}a^{-4}-3z^{2}a^{-4}-a^{-4}+9z^{7}a^{-3}-14z^{5}a^{-3}+10z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}+4z^{6}a^{-2}-4z^{4}a^{-2}+z^{5}a^{-1}-z^{3}a^{-1}$ (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 22                       1 -1 20                     3   3 18                   7 1   -6 16                 9 3     6 14               13 7       -6 12             13 9         4 10           12 13           1 8         11 13             -2 6       6 13               7 4     4 10                 -6 2   1 7                   6 0   3                     -3 -2 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=2$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a232 L11a234