L10n69

L10n68

L10n70

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Link Presentations

[edit Notes on L10n69's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X15,1,16,4 X19,10,20,5
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, 4, -5, -9, 7, -8, 6, -10, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2uvw^{2}-2uvw-uw^{2}+2uw-2v^{2}w+v^{2}+2vw-2v}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{-9}-2q^{-8}+3q^{-7}-4q^{-6}+5q^{-5}-4q^{-4}+5q^{-3}-2q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{2}a^{8}+a^{8}-z^{4}a^{6}-2z^{2}a^{6}+a^{6}z^{-2}-a^{6}-z^{4}a^{4}-2z^{2}a^{4}-2a^{4}z^{-2}-3a^{4}+2z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-4a^{10}z^{4}+4a^{10}z^{2}-a^{10}+2a^{9}z^{7}-8a^{9}z^{5}+8a^{9}z^{3}-2a^{9}z+a^{8}z^{8}-a^{8}z^{6}-6a^{8}z^{4}+6a^{8}z^{2}-a^{8}+4a^{7}z^{7}-14a^{7}z^{5}+14a^{7}z^{3}-6a^{7}z+a^{6}z^{8}-7a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+5a^{5}z^{3}-2a^{5}z^{-1}+2a^{4}z^{6}-5a^{4}z^{4}+9a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+a^{3}z^{5}-a^{3}z^{3}+4a^{3}z-2a^{3}z^{-1}+3a^{2}z^{2}+a^{2}z^{-2}-4a^{2}$ (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 χ -1                 2 2 -3               2 2 0 -5             3     3 -7           2 3     1 -9         3 2       1 -11       1 2         1 -13     2 3           -1 -15   1 2             1 -17   1               -1 -19 1                 1

Integral Khovanov Homology (db, data source)

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

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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