L11n293

L11n292

L11n294

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Link Presentations

[edit Notes on L11n293's Link Presentations]

Planar diagram presentation	X6172 X11,18,12,19 X8493 X2,16,3,15 X16,7,17,8 X9,11,10,22 X4,17,1,18 X19,5,20,10 X5,12,6,13 X21,15,22,14 X13,21,14,20
Gauss code	{1, -4, 3, -7}, {-9, -1, 5, -3, -6, 8}, {-2, 9, -11, 10, 4, -5, 7, 2, -8, 11, -10, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}+4q^{4}-4q^{3}+4q^{2}-2q+3+q^{-1}-q^{-2}+2q^{-3}-q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}-z^{6}a^{-2}-a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}-8z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-a^{2}-3a^{-2}+z^{6}+6z^{4}+9z^{2}-2z^{-2}+3$ (db)
Kauffman polynomial	$z^{6}a^{-6}-3z^{4}a^{-6}+z^{2}a^{-6}+3z^{7}a^{-5}-11z^{5}a^{-5}+7z^{3}a^{-5}-2za^{-5}+3z^{8}a^{-4}-12z^{6}a^{-4}+10z^{4}a^{-4}-6z^{2}a^{-4}+2a^{-4}+z^{9}a^{-3}+a^{3}z^{7}-z^{7}a^{-3}-5a^{3}z^{5}-13z^{5}a^{-3}+5a^{3}z^{3}+20z^{3}a^{-3}-a^{3}z-7za^{-3}+2a^{2}z^{8}+5z^{8}a^{-2}-12a^{2}z^{6}-30z^{6}a^{-2}+19a^{2}z^{4}+50z^{4}a^{-2}-13a^{2}z^{2}-30z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+6a^{-2}+az^{9}+2z^{9}a^{-1}-5az^{7}-10z^{7}a^{-1}+3z^{5}a^{-1}+11az^{3}+19z^{3}a^{-1}-5az-9za^{-1}-2az^{-1}-2a^{-1}z^{-1}+4z^{8}-29z^{6}+56z^{4}-36z^{2}+2z^{-2}+7$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   2 1   1 7               1 3 2     0 5             1 3 2       0 3           2 3 3         2 1         2 6 3           1 -1       1 2 5             4 -3     1 2 1               0 -5   1 1 1                 1 -7   1                     1 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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