L11n157

L11n156

L11n158

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n157 at Knotilus!

Link Presentations

[edit Notes on L11n157's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X17,15,18,14 X7,17,8,16 X15,7,16,22 X13,19,14,18 X6,20,1,19 X20,12,21,11 X12,6,13,5 X4,21,5,22
Gauss code	{1, -2, 3, -11, 10, -8}, {-5, -1, 2, -3, 9, -10, -7, 4, -6, 5, -4, 7, 8, -9, 11, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-2t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+7q^{7/2}-{\frac {1}{q^{7/2}}}-10q^{5/2}+{\frac {3}{q^{5/2}}}+10q^{3/2}-{\frac {6}{q^{3/2}}}+2q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}+3az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+3az-5za^{-1}+za^{-5}+az^{-1}-2a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-6}-6z^{2}a^{-6}+2a^{-6}+z^{7}a^{-5}+z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-4z^{6}a^{-4}+10z^{4}a^{-4}-12z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}+2z^{7}a^{-3}+a^{3}z^{5}-5z^{5}a^{-3}-2a^{3}z^{3}+6z^{3}a^{-3}+a^{3}z-2a^{-3}z^{-1}+5z^{8}a^{-2}+3a^{2}z^{6}-9z^{6}a^{-2}-6a^{2}z^{4}+9z^{4}a^{-2}+2a^{2}z^{2}-5z^{2}a^{-2}+3a^{-2}+z^{9}a^{-1}+4az^{7}+5z^{7}a^{-1}-7az^{5}-14z^{5}a^{-1}+3az^{3}+14z^{3}a^{-1}-3az-5za^{-1}+az^{-1}+3z^{8}-2z^{6}-4z^{4}+3z^{2}-1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   2 -2 10                 2   2 8               5 2   -3 6             5 2     3 4           5 5       0 2         6 5         1 0       3 6           3 -2     3 5             -2 -4   1 4               3 -6   2                 -2 -8 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n156 L11n158