L10n91

L10n90

L10n92

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Link Presentations

[edit Notes on L10n91's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X4,11,1,12
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(3)-1)(t(3)+1)(-2t(1)t(2)+t(1)t(3)t(2)-2t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-q^{-1}+q^{-2}-q^{-3}+q^{-5}+q^{-6}+2q^{-7}-q^{-8}+q^{-9}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}-2a^{8}z^{-2}-4a^{8}+z^{4}a^{6}+6z^{2}a^{6}+a^{6}z^{-2}+6a^{6}-z^{6}a^{4}-5z^{4}a^{4}-6z^{2}a^{4}-4a^{4}+z^{4}a^{2}+4z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{6}-5a^{10}z^{4}+6a^{10}z^{2}+a^{10}z^{-2}-4a^{10}+a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+3a^{8}z^{4}+6a^{8}z^{2}+2a^{8}z^{-2}-5a^{8}+2a^{7}z^{7}-12a^{7}z^{5}+16a^{7}z^{3}-4a^{7}z-2a^{7}z^{-1}+a^{6}z^{8}-5a^{6}z^{6}+2a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+2a^{5}z^{7}-13a^{5}z^{5}+22a^{5}z^{3}-12a^{5}z+2a^{4}z^{6}-11a^{4}z^{4}+12a^{4}z^{2}-2a^{4}+a^{3}z^{7}-5a^{3}z^{5}+6a^{3}z^{3}-4a^{3}z+a^{2}z^{6}-5a^{2}z^{4}+6a^{2}z^{2}-2a^{2}$ (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 1 2 χ 1                     1 1 -1                       0 -3               1 2 1   0 -5               1 1     0 -7           2 2 1       -1 -9         1 1 1         1 -11       1 4 1           2 -13     1 1 3             3 -15     1                 1 -17 1 1                   0 -19 1                     1

Integral Khovanov Homology (db, data source)

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

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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