L11a322

L11a321

L11a323

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a322 at Knotilus!

Link Presentations

[edit Notes on L11a322's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-2t(1)t(2)^{3}+t(2)^{3}-2t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-2t(2)^{2}+t(1)^{2}t(2)-2t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {4}{q^{9/2}}}-8q^{7/2}+{\frac {9}{q^{7/2}}}+13q^{5/2}-{\frac {15}{q^{5/2}}}-19q^{3/2}+{\frac {19}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+21{\sqrt {q}}-{\frac {22}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+3az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-a^{3}z+2az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-9az^{9}-15z^{9}a^{-1}-6z^{9}a^{-3}-14a^{2}z^{8}-2z^{8}a^{-2}-4z^{8}a^{-4}-12z^{8}-14a^{3}z^{7}+6az^{7}+41z^{7}a^{-1}+20z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+21a^{2}z^{6}+28z^{6}a^{-2}+14z^{6}a^{-4}+44z^{6}-4a^{5}z^{5}+20a^{3}z^{5}+17az^{5}-30z^{5}a^{-1}-20z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-7a^{2}z^{4}-31z^{4}a^{-2}-14z^{4}a^{-4}-32z^{4}+a^{5}z^{3}-11a^{3}z^{3}-14az^{3}+6z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}+2a^{3}z+4az+2za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 12                       1 1 10                     3   -3 8                   5 1   4 6                 8 3     -5 4               11 5       6 2             10 8         -2 0           12 11           1 -2         9 12             3 -4       6 10               -4 -6     3 9                 6 -8   1 6                   -5 -10   3                     3 -12 1                       -1

Integral Khovanov Homology (db, data source)

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

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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