L11n405

L11n404

L11n406

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n405 at Knotilus!

Link Presentations

[edit Notes on L11n405's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X22,12,19,11 X10,4,11,3 X5,21,6,20 X21,5,22,18 X12,20,13,19 X2,14,3,13
Gauss code	{1, -11, 7, -5}, {10, 8, -9, -6}, {-8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)^{2}\left(w^{2}+1\right)(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{-5}+3q^{4}+3q^{-4}-4q^{3}-6q^{-3}+8q^{2}+8q^{-2}-9q-10q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-5z^{4}a^{-2}+13z^{4}-5a^{2}z^{2}-10z^{2}a^{-2}+z^{2}a^{-4}+14z^{2}-3a^{2}-11a^{-2}+3a^{-4}+11-a^{2}z^{-2}-5a^{-2}z^{-2}+2a^{-4}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-2a^{5}z^{3}+a^{5}z+3a^{4}z^{6}-6a^{4}z^{4}+6z^{4}a^{-4}+a^{4}z^{2}-17z^{2}a^{-4}-2a^{-4}z^{-2}+10a^{-4}+4a^{3}z^{7}+3z^{7}a^{-3}-7a^{3}z^{5}-9z^{5}a^{-3}+a^{3}z^{3}+14z^{3}a^{-3}-2a^{3}z-15za^{-3}+a^{3}z^{-1}+5a^{-3}z^{-1}+4a^{2}z^{8}+5z^{8}a^{-2}-9a^{2}z^{6}-22z^{6}a^{-2}+10a^{2}z^{4}+46z^{4}a^{-2}-9a^{2}z^{2}-45z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}+4a^{2}+22a^{-2}+2az^{9}+2z^{9}a^{-1}-z^{7}a^{-1}-11az^{5}-12z^{5}a^{-1}+23az^{3}+34z^{3}a^{-1}-17az-29za^{-1}+5az^{-1}+9a^{-1}z^{-1}+9z^{8}-34z^{6}+56z^{4}-38z^{2}-4z^{-2}+17$ (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 χ 9                   3 3 7                 3 2 -1 5               5 1   4 3             4 3     -1 1           8 5       3 -1         4 6         2 -3       4 6           -2 -5     2 4             2 -7   1 4               -3 -9   2                 2 -11 1                   -1

Integral Khovanov Homology (db, data source)

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

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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