L11n358

L11n357

L11n359

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Link Presentations

[edit Notes on L11n358's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,19,12,18 X7,16,8,17 X15,8,16,9 X17,15,18,22 X21,13,22,12 X13,21,14,20 X19,5,20,14 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(2)t(3)^{3}-t(3)^{3}+t(2)^{2}t(3)^{2}-2t(2)t(3)^{2}+t(1)t(3)-2t(1)t(2)t(3)-t(1)t(2)^{2}+t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{3}+3q^{2}-4q+7-6q^{-1}+7q^{-2}-5q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-2a^{4}z^{2}+a^{4}z^{-2}-a^{4}+a^{2}z^{4}-2a^{2}z^{-2}-z^{2}a^{-2}-2a^{2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+2a^{4}z^{8}+5a^{2}z^{8}+3z^{8}+2a^{5}z^{7}+2z^{7}a^{-1}+a^{6}z^{6}-6a^{4}z^{6}-22a^{2}z^{6}-15z^{6}-7a^{5}z^{5}-10a^{3}z^{5}-11az^{5}-8z^{5}a^{-1}-4a^{6}z^{4}+4a^{4}z^{4}+37a^{2}z^{4}+3z^{4}a^{-2}+32z^{4}+5a^{5}z^{3}+15a^{3}z^{3}+24az^{3}+15z^{3}a^{-1}+z^{3}a^{-3}+4a^{6}z^{2}-5a^{4}z^{2}-30a^{2}z^{2}-5z^{2}a^{-2}-26z^{2}-a^{5}z-10a^{3}z-14az-7za^{-1}-2za^{-3}-a^{6}+4a^{4}+12a^{2}+2a^{-2}+10+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   1 -1 5                 2   2 3               3 2   -1 1             4 1     3 -1           3 4       1 -3         4 3         1 -5       2 4           2 -7     2 3             -1 -9   1 3               2 -11   1                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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