L11a452

L11a451

L11a453

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a452 at Knotilus!

Link Presentations

[edit Notes on L11a452's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-2t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+9q^{3}-10q^{2}+12q-11+10q^{-1}-6q^{-2}+5q^{-3}-2q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}+a^{4}z^{2}-2z^{2}a^{-4}+a^{4}z^{-2}+2a^{4}-a^{-4}+z^{6}a^{-2}-2a^{2}z^{4}+3z^{4}a^{-2}-6a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}-5a^{2}+2a^{-2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+a^{4}z^{8}+a^{2}z^{8}+7z^{8}a^{-2}+7z^{8}-10a^{3}z^{7}-24az^{7}-5z^{7}a^{-1}+9z^{7}a^{-3}-6a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}+9z^{6}a^{-4}-40z^{6}+14a^{3}z^{5}+24az^{5}-12z^{5}a^{-1}-16z^{5}a^{-3}+6z^{5}a^{-5}+13a^{4}z^{4}+52a^{2}z^{4}-14z^{4}a^{-4}+3z^{4}a^{-6}+56z^{4}-2a^{3}z^{3}+az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}-13a^{4}z^{2}-39a^{2}z^{2}+z^{2}a^{-2}+7z^{2}a^{-4}-32z^{2}-5a^{3}z-8az-3za^{-1}+za^{-3}+za^{-5}+6a^{4}+13a^{2}-a^{-4}+9+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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 13                       1 -1 11                     2   2 9                   4 1   -3 7                 5 2     3 5               5 4       -1 3             7 5         2 1           7 8           1 -1         3 4             -1 -3       3 7               4 -5     2 3                 -1 -7   1 4                   3 -9   1                     -1 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a451 L11a453