L11n449

L11n448

L11n450

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n449 at Knotilus!

Link Presentations

[edit Notes on L11n449's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X13,20,14,21 X16,12,17,11 X19,12,20,13 X8,16,5,15 X14,8,15,7 X22,17,19,18 X18,21,9,22 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, 9, -8}, {11, -2, 4, 5, -3, -7, 6, -4, 8, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(4)-1)\left(-t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-2t(2)t(4)^{2}+t(4)^{2}+t(1)t(4)-t(2)t(3)t(4)+2t(1)t(3)-t(1)t(2)t(3)+t(2)t(3)-t(3)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {6}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z^{-1}-3a^{7}z^{-3}-3a^{7}z-5a^{7}z^{-1}+3a^{5}z^{3}+3a^{5}z^{-3}+6a^{5}z+6a^{5}z^{-1}-a^{3}z^{5}-2a^{3}z^{3}-a^{3}z^{-3}-2a^{3}z-a^{3}z^{-1}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-6a^{9}z^{5}+13a^{9}z^{3}-a^{9}z^{-3}-13a^{9}z+6a^{9}z^{-1}+a^{8}z^{8}-2a^{8}z^{6}-6a^{8}z^{4}+17a^{8}z^{2}+3a^{8}z^{-2}-13a^{8}+a^{7}z^{9}-14a^{7}z^{5}+29a^{7}z^{3}-3a^{7}z^{-3}-27a^{7}z+14a^{7}z^{-1}+5a^{6}z^{8}-14a^{6}z^{6}-3a^{6}z^{4}+28a^{6}z^{2}+6a^{6}z^{-2}-24a^{6}+a^{5}z^{9}+5a^{5}z^{7}-26a^{5}z^{5}+30a^{5}z^{3}-3a^{5}z^{-3}-22a^{5}z+12a^{5}z^{-1}+4a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{2}+3a^{4}z^{-2}-11a^{4}+6a^{3}z^{7}-18a^{3}z^{5}+19a^{3}z^{3}-a^{3}z^{-3}-8a^{3}z+3a^{3}z^{-1}+3a^{2}z^{6}-3a^{2}z^{4}+a^{2}z^{2}+a^{2}+5az^{3}-az^{-1}+z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   1 -1 0                 4   4 -2               5 4   -1 -4             3 2 1   2 -6           4 5       1 -8         5 3         2 -10       4 8           4 -12     1 1             0 -14     4               4 -16 1 1                 0 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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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