L11n429

L11n428

L11n430

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n429 at Knotilus!

Link Presentations

[edit Notes on L11n429's Link Presentations]

Planar diagram presentation	X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X19,4,20,5 X2,21,3,22
Gauss code	{1, -11, -7, 10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{3}-2u^{2}v^{2}w^{2}+u^{2}v^{2}w-u^{2}vw^{3}+2u^{2}vw^{2}+uv^{2}w^{2}-uv^{2}w-uvw^{2}+uvw+uw^{2}-uw-2vw+v-w^{2}+2w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q+3-4q^{-1}+6q^{-2}-6q^{-3}+7q^{-4}-5q^{-5}+5q^{-6}-2q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}z^{-2}+a^{8}-a^{6}z^{6}-5a^{6}z^{4}-7a^{6}z^{2}-2a^{6}z^{-2}-5a^{6}+a^{4}z^{8}+6a^{4}z^{6}+12a^{4}z^{4}+11a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-a^{2}z^{6}-4a^{2}z^{4}-3a^{2}z^{2}$ (db)
Kauffman polynomial	$a^{10}z^{2}+2a^{9}z^{3}+5a^{8}z^{4}-9a^{8}z^{2}-a^{8}z^{-2}+6a^{8}+2a^{7}z^{7}-5a^{7}z^{5}+6a^{7}z^{3}-6a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-18a^{6}z^{6}+31a^{6}z^{4}-29a^{6}z^{2}-2a^{6}z^{-2}+12a^{6}+2a^{5}z^{9}-5a^{5}z^{7}-3a^{5}z^{5}+10a^{5}z^{3}-8a^{5}z+2a^{5}z^{-1}+7a^{4}z^{8}-32a^{4}z^{6}+44a^{4}z^{4}-26a^{4}z^{2}-a^{4}z^{-2}+8a^{4}+2a^{3}z^{9}-6a^{3}z^{7}-2a^{3}z^{5}+10a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-14a^{2}z^{6}+18a^{2}z^{4}-7a^{2}z^{2}+a^{2}+az^{7}-4az^{5}+4az^{3}-az$ (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 χ 3                   1 -1 1                 2   2 -1               2 1   -1 -3             4 2     2 -5           3 3       0 -7         4 3         1 -9       2 4           2 -11     3 3             0 -13     3               3 -15 1 2                 -1 -17 1                   1

Integral Khovanov Homology (db, data source)

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

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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