L11n359

L11n358

L11n360

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n359 at Knotilus!

Link Presentations

[edit Notes on L11n359's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X17,9,18,8 X7,17,8,16 X15,5,16,14 X19,15,20,22 X13,20,14,21 X21,12,22,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -4, 3, -7, 8, -9, 7}, {10, -1, -5, 4, 11, -2, -3, 9, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(2)^{2}t(3)^{3}-t(2)t(3)^{3}-t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(2)^{2}t(3)-t(1)t(2)t(3)+t(2)t(3)+t(1)-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{5}+q^{-5}+2q^{4}-q^{-4}-2q^{3}+2q^{-3}+3q^{2}-2q+2$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{6}-2a^{2}z^{4}+6z^{4}+a^{4}z^{2}-9a^{2}z^{2}-2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+3a^{4}-8a^{2}-a^{-2}+6+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+a^{4}z^{8}+3a^{2}z^{8}+2z^{8}-6a^{3}z^{7}-6az^{7}+z^{7}a^{-1}+z^{7}a^{-3}-7a^{4}z^{6}-22a^{2}z^{6}+2z^{6}a^{-2}+2z^{6}a^{-4}-15z^{6}+8a^{3}z^{5}+5az^{5}-6z^{5}a^{-1}-2z^{5}a^{-3}+z^{5}a^{-5}+16a^{4}z^{4}+48a^{2}z^{4}-7z^{4}a^{-2}-7z^{4}a^{-4}+32z^{4}+2a^{3}z^{3}+10az^{3}+9z^{3}a^{-1}-2z^{3}a^{-3}-3z^{3}a^{-5}-16a^{4}z^{2}-40a^{2}z^{2}+5z^{2}a^{-2}+5z^{2}a^{-4}-24z^{2}-7a^{3}z-10az-3za^{-1}+za^{-3}+za^{-5}+7a^{4}+14a^{2}-a^{-2}-a^{-4}+8+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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 11                       1 -1 9                     1   1 7                   1 1   0 5               1 3 1     1 3             1 2 2       1 1           1 3 2         0 -1         1 2 3           2 -3       1 2 1             0 -5     1 1 2               2 -7   1 2                   1 -9                         0 -11 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11n358 L11n360