L11n430

L11n429

L11n431

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n430 at Knotilus!

Link Presentations

[edit Notes on L11n430'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 X4,20,5,19 X21,3,22,2
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 $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(u^{2}w^{2}-uvw+uv-uw^{2}+uw-v\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+3q^{2}-5q+8-7q^{-1}+9q^{-2}-6q^{-3}+5q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{6}-4a^{2}z^{4}-6a^{2}z^{2}-z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+2z^{4}+5z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-10a^{5}z^{5}+6a^{5}z^{3}+4a^{4}z^{8}-15a^{4}z^{6}+16a^{4}z^{4}-9a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-3a^{3}z^{5}+5a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-za^{-3}+2a^{3}z^{-1}+7a^{2}z^{8}-29a^{2}z^{6}+43a^{2}z^{4}+3z^{4}a^{-2}-27a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+2a^{-2}+2az^{9}-6az^{7}+z^{7}a^{-1}+6az^{5}-z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-4az-3za^{-1}+2az^{-1}+3z^{8}-13z^{6}+27z^{4}-21z^{2}-z^{-2}+7$ (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 7                   1 -1 5                 2   2 3               3 1   -2 1             5 2     3 -1           4 5       1 -3         5 3         2 -5       3 6           3 -7     2 3             -1 -9   1 3               2 -11   2                 -2 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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