L11n436

L11n435

L11n437

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n436 at Knotilus!
	Brunnian link

Link Presentations

[edit Notes on L11n436's Link Presentations]

Planar diagram presentation	X8192 X18,8,19,7 X5,16,6,17 X15,10,16,11 X3,15,4,14 X11,5,12,4 X2,20,3,19 X20,9,21,10 X13,7,14,12 X17,22,18,13 X6,21,1,22
Gauss code	{1, -7, -5, 6, -3, -11}, {2, -1, 8, 4, -6, 9}, {-9, 5, -4, 3, -10, -2, 7, -8, 11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(w+1)^{2}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{4}+2q^{3}-q^{2}+2q+1+q^{-2}-q^{-3}+2q^{-4}-q^{-5}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-z^{4}a^{-2}-a^{2}z^{2}-3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+z^{6}+5z^{4}+5z^{2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}-3a^{5}z^{3}+2a^{4}z^{6}-8a^{4}z^{4}+4a^{4}z^{2}+a^{3}z^{7}+z^{7}a^{-3}-5a^{3}z^{5}-5z^{5}a^{-3}+4a^{3}z^{3}+5z^{3}a^{-3}+a^{2}z^{8}+2z^{8}a^{-2}-6a^{2}z^{6}-12z^{6}a^{-2}+8a^{2}z^{4}+20z^{4}a^{-2}-4a^{2}z^{2}-12z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+az^{9}+z^{9}a^{-1}-5az^{7}-5z^{7}a^{-1}+2az^{5}+3z^{5}a^{-1}+6az^{3}+4z^{3}a^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-20z^{6}+36z^{4}-20z^{2}+2z^{-2}+1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                     1 -1 7                   1   1 5               1 1 1   1 3               2 1     1 1           4 2 1       3 -1         2 5 2         1 -3       1 2 2           1 -5     1 2 1             0 -7   1 1 1               1 -9   1                   1 -11 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

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