L11n448

L11n447

L11n449

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n448 at Knotilus!

Link Presentations

[edit Notes on L11n448'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 X17,19,18,22 X21,9,22,18 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 $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+4q^{5/2}-{\frac {4}{q^{5/2}}}-5q^{3/2}+{\frac {3}{q^{3/2}}}+4{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+a^{5}z^{-1}-3a^{3}z^{-3}+z^{3}a^{-3}-2a^{3}z-4a^{3}z^{-1}+za^{-3}-z^{5}a^{-1}+2az^{3}+3az^{-3}-3z^{3}a^{-1}-a^{-1}z^{-3}+4az+5az^{-1}-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{3}-a^{5}z^{-3}-3a^{5}z+3a^{5}z^{-1}+z^{6}a^{-4}-3z^{4}a^{-4}+3a^{4}z^{2}+z^{2}a^{-4}+3a^{4}z^{-2}-6a^{4}+3z^{7}a^{-3}+a^{3}z^{5}-11z^{5}a^{-3}+8z^{3}a^{-3}-3a^{3}z^{-3}-5a^{3}z-3za^{-3}+6a^{3}z^{-1}+a^{2}z^{8}+3z^{8}a^{-2}-5a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+7z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}+6a^{2}z^{-2}-11a^{2}+az^{9}+z^{9}a^{-1}-3az^{7}-az^{5}-13z^{5}a^{-1}+10az^{3}+19z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-7az-8za^{-1}+6az^{-1}+3a^{-1}z^{-1}+4z^{8}-17z^{6}+19z^{4}-2z^{2}+3z^{-2}-6$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               2 1   -1 4             3 2     1 2         2 3 2       1 0       1 7 3         3 -2       3 7           4 -4     2 1             1 -6 1   3               4 -8 2 1                 1 -10 1                   1

Integral Khovanov Homology (db, data source)

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

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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