L11n346

L11n345

L11n347

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Link Presentations

[edit Notes on L11n346's Link Presentations]

Planar diagram presentation	X6172 X3,15,4,14 X11,20,12,21 X7,18,8,19 X17,22,18,13 X9,17,10,16 X15,11,16,10 X19,12,20,5 X21,8,22,9 X2536 X13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -6, 7, -3, 8}, {-11, 2, -7, 6, -5, 4, -8, 3, -9, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(2)^{3}-t(1)t(3)t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-3t(1)t(3)^{2}t(2)^{2}+3t(3)^{2}t(2)^{2}-2t(1)t(2)^{2}+4t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}-t(1)t(3)^{3}t(2)+2t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-4t(3)^{2}t(2)+t(1)t(2)-3t(1)t(3)t(2)+3t(3)t(2)-t(2)-t(3)^{3}+t(3)^{2}}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+7q-10+13q^{-1}-12q^{-2}+13q^{-3}-9q^{-4}+6q^{-5}-2q^{-6}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}-z^{4}a^{4}+z^{2}a^{4}+a^{4}z^{-2}+3a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2a^{2}z^{-2}-3a^{2}-2z^{4}-3z^{2}+z^{-2}+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$2a^{3}z^{9}+2az^{9}+6a^{4}z^{8}+10a^{2}z^{8}+4z^{8}+5a^{5}z^{7}+6a^{3}z^{7}+4az^{7}+3z^{7}a^{-1}+a^{6}z^{6}-15a^{4}z^{6}-25a^{2}z^{6}+z^{6}a^{-2}-8z^{6}-9a^{5}z^{5}-22a^{3}z^{5}-21az^{5}-8z^{5}a^{-1}+6a^{6}z^{4}+27a^{4}z^{4}+25a^{2}z^{4}-3z^{4}a^{-2}+z^{4}+3a^{7}z^{3}+15a^{5}z^{3}+23a^{3}z^{3}+17az^{3}+6z^{3}a^{-1}-6a^{6}z^{2}-26a^{4}z^{2}-23a^{2}z^{2}+3z^{2}a^{-2}-2a^{7}z-7a^{5}z-12a^{3}z-8az-za^{-1}+3a^{6}+11a^{4}+11a^{2}-a^{-2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               5 1   4 1             5 2     -3 -1           8 5       3 -3         7 8         1 -5       6 5           1 -7     3 7             4 -9   3 6               -3 -11   4                 4 -13 2                   -2

Integral Khovanov Homology (db, data source)

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

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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