L11a434

L11a433

L11a435

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a434 at Knotilus!
	Brunnian link

Link Presentations

[edit Notes on L11a434's Link Presentations]

Planar diagram presentation	X6172 X16,12,17,11 X8493 X2,18,3,17 X14,6,15,5 X18,7,19,8 X12,16,5,15 X20,14,21,13 X22,9,13,10 X10,21,11,22 X4,19,1,20
Gauss code	{1, -4, 3, -11}, {5, -1, 6, -3, 9, -10, 2, -7}, {8, -5, 7, -2, 4, -6, 11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)(w-1)(v+w-1)(vw-v-w)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-4q^{5}+8q^{4}-14q^{3}+21q^{2}-22q+24-20q^{-1}+16q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-z^{6}+2a^{2}z^{4}-2z^{4}a^{-2}+z^{4}a^{-4}-a^{4}z^{2}+a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+7az^{9}+13z^{9}a^{-1}+6z^{9}a^{-3}+10a^{2}z^{8}+14z^{8}a^{-2}+7z^{8}a^{-4}+17z^{8}+8a^{3}z^{7}-az^{7}-20z^{7}a^{-1}-7z^{7}a^{-3}+4z^{7}a^{-5}+4a^{4}z^{6}-14a^{2}z^{6}-46z^{6}a^{-2}-18z^{6}a^{-4}+z^{6}a^{-6}-45z^{6}+a^{5}z^{5}-11a^{3}z^{5}-12az^{5}+3z^{5}a^{-1}-7z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+8a^{2}z^{4}+48z^{4}a^{-2}+16z^{4}a^{-4}-2z^{4}a^{-6}+43z^{4}-a^{5}z^{3}+5a^{3}z^{3}+7az^{3}+5z^{3}a^{-1}+10z^{3}a^{-3}+6z^{3}a^{-5}+2a^{4}z^{2}-4a^{2}z^{2}-20z^{2}a^{-2}-6z^{2}a^{-4}-20z^{2}+1-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 9 3     -6 5               12 5       7 3             10 9         -1 1           14 12           2 -1         10 14             4 -3       6 10               -4 -5     3 10                 7 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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