L11a353

L11a352

L11a354

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a353 at Knotilus!

Link Presentations

[edit Notes on L11a353's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X16,6,17,5 X22,8,11,7 X20,15,21,16 X14,21,15,22 X6,14,7,13 X4,20,5,19 X18,9,19,10 X2,11,3,12 X10,17,1,18
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-3u^{2}v+2u^{2}-uv^{2}+5uv-u+2v^{2}-3v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-17q^{5/2}+21q^{3/2}-25{\sqrt {q}}+{\frac {24}{\sqrt {q}}}-{\frac {21}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+7az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-3a^{3}z+6az-7za^{-1}+3za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+4z^{5}a^{-5}+3a^{5}z^{3}-a^{5}z+4a^{4}z^{8}-13a^{4}z^{6}+10z^{6}a^{-4}+14a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+3z^{2}a^{-4}+6a^{3}z^{9}-16a^{3}z^{7}+17z^{7}a^{-3}+9a^{3}z^{5}-25z^{5}a^{-3}+4a^{3}z^{3}+16z^{3}a^{-3}-2a^{3}z-5za^{-3}+3a^{2}z^{10}+7a^{2}z^{8}+18z^{8}a^{-2}-42a^{2}z^{6}-29z^{6}a^{-2}+43a^{2}z^{4}+12z^{4}a^{-2}-11a^{2}z^{2}-z^{2}a^{-2}+17az^{9}+11z^{9}a^{-1}-38az^{7}-4z^{7}a^{-1}+9az^{5}-32z^{5}a^{-1}+14az^{3}+29z^{3}a^{-1}-6az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{10}+21z^{8}-68z^{6}+49z^{4}-10z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   7 1   -6 6                 10 3     7 4               11 7       -4 2             14 10         4 0           12 13           1 -2         9 12             -3 -4       6 12               6 -6     3 9                 -6 -8   1 6                   5 -10   3                     -3 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a352 L11a354