L11n446

L11n445

L11n447

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n446 at Knotilus!

Link Presentations

[edit Notes on L11n446's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X18,12,19,11 X22,19,17,20 X16,21,9,22 X20,15,21,16 X12,18,13,17 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {9, -5, 6, -8, 7, -6}, {-11, 2, 5, -9, -4, 3, 8, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(w-1)(x-1)(u-x)(x-v)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-3q^{7/2}-{\frac {2}{q^{7/2}}}+q^{5/2}+{\frac {2}{q^{5/2}}}-2q^{3/2}-{\frac {2}{q^{3/2}}}-q^{11/2}-2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{-5}z^{-3}-za^{-5}-2a^{-5}z^{-1}-a^{3}z^{3}+2z^{3}a^{-3}+3a^{-3}z^{-3}-2a^{3}z+7za^{-3}-a^{3}z^{-1}+8a^{-3}z^{-1}+az^{5}-z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+8az-12za^{-1}+6az^{-1}-11a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}a^{-2}-z^{8}a^{-4}-z^{8}-2a^{3}z^{7}-3az^{7}-4z^{7}a^{-1}-4z^{7}a^{-3}-z^{7}a^{-5}-a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+4z^{6}a^{-4}+4z^{6}+9a^{3}z^{5}+20az^{5}+29z^{5}a^{-1}+24z^{5}a^{-3}+6z^{5}a^{-5}+4a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}+8z^{4}-9a^{3}z^{3}-36az^{3}-57z^{3}a^{-1}-42z^{3}a^{-3}-12z^{3}a^{-5}-3a^{4}z^{2}-14a^{2}z^{2}-24z^{2}a^{-2}-9z^{2}a^{-4}-26z^{2}+6a^{3}z+27az+44za^{-1}+34za^{-3}+11za^{-5}+a^{4}+6a^{2}+21a^{-2}+9a^{-4}+18-2a^{3}z^{-1}-11az^{-1}-18a^{-1}z^{-1}-14a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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 12                       1 1 10                         0 8                   3 1   2 6               1 1 2     2 4               2 1       1 2           5 2 1         4 0         2 7 2           3 -2       1 1 3             3 -4     1 2 1               0 -6   1 1                   0 -8   1                     1 -10 1                       -1

Integral Khovanov Homology (db, data source)

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

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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