L11a446

L11a445

L11a447

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a446 at Knotilus!

Link Presentations

[edit Notes on L11a446's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X18,9,19,10 X16,8,17,7 X22,17,13,18 X20,11,21,12 X8,14,9,13 X10,19,11,20 X12,21,5,22 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 3, -8, 6, -9}, {7, -2, 11, -4, 5, -3, 8, -6, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-2uv^{2}w^{3}+4uv^{2}w^{2}-2uv^{2}w+uvw^{3}-3uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+2vw^{2}-4vw+2v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+6q-8+13q^{-1}-14q^{-2}+15q^{-3}-12q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+13a^{4}z^{2}-16a^{2}z^{2}+5z^{2}-2a^{6}+7a^{4}-9a^{2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}-a^{8}+4a^{7}z^{7}-5a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+4a^{6}z^{8}-4a^{6}z^{6}-2a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{9}-2a^{5}z^{7}-3a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+a^{4}z^{10}+7a^{4}z^{8}-29a^{4}z^{6}+47a^{4}z^{4}-31a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+7a^{3}z^{9}-20a^{3}z^{7}+19a^{3}z^{5}+4a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}-41a^{2}z^{6}+z^{6}a^{-2}+67a^{2}z^{4}-3z^{4}a^{-2}-44a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4az^{9}-11az^{7}+3z^{7}a^{-1}+7az^{5}-9z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-5az+2az^{-1}+5z^{8}-18z^{6}+21z^{4}-14z^{2}-z^{-2}+6$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   4 1   3 1                 4 2     -2 -1               9 4       5 -3             8 7         -1 -5           7 6           1 -7         5 8             3 -9       5 7               -2 -11     2 6                 4 -13   1 4                   -3 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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