L11n2

L11n1

L11n3

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Link L11n2.

A graph, L11n2.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n2's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X15,1,16,4 X5,10,6,11 X3849 X11,20,12,21 X17,5,18,22 X21,19,22,18 X19,12,20,13 X9,16,10,17 X2,14,3,13
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -10, 4, -6, 9, 11, -2, -3, 10, -7, 8, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-4v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {7}{q^{7/2}}}-{\frac {9}{q^{5/2}}}-2q^{3/2}+{\frac {7}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+4{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+4a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-6a^{3}z-3a^{3}z^{-1}+3az^{3}+5az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-2a^{6}z^{8}-5a^{4}z^{8}-3a^{2}z^{8}-2a^{7}z^{7}-a^{5}z^{7}-2a^{3}z^{7}-3az^{7}-a^{8}z^{6}+5a^{6}z^{6}+17a^{4}z^{6}+10a^{2}z^{6}-z^{6}+7a^{7}z^{5}+12a^{5}z^{5}+14a^{3}z^{5}+9az^{5}+4a^{8}z^{4}-20a^{4}z^{4}-18a^{2}z^{4}-2z^{4}-6a^{7}z^{3}-16a^{5}z^{3}-24a^{3}z^{3}-17az^{3}-3z^{3}a^{-1}-4a^{8}z^{2}-3a^{6}z^{2}+8a^{4}z^{2}+11a^{2}z^{2}+4z^{2}+3a^{7}z+10a^{5}z+14a^{3}z+11az+4za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 4                   2 2 2                 2   -2 0               4 2   2 -2             5 4     -1 -4           4 2       2 -6         3 5         2 -8       3 4           -1 -10     1 3             2 -12   1 3               -2 -14   1                 1 -16 1                   -1

Integral Khovanov Homology (db, data source)

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

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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