L11n459

L11n458

L11n459

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Link L11n459.

A graph, L11n459.

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

Link Presentations

[edit Notes on L11n459's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {11, -2, -5, 7, -9, 8, -7, 6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(wx-1)(uwx+u(-w)-ux+2u-2vwx+vw+vx-v)}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-2q^{9/2}-q^{5/2}-{\frac {1}{q^{5/2}}}-2q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+q^{11/2}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{3}a^{-5}-a^{-5}z^{-3}-2za^{-5}-2a^{-5}z^{-1}+z^{3}a^{-3}+3a^{-3}z^{-3}+6za^{-3}+7a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-8za^{-1}+3az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-3z^{8}a^{-4}-z^{8}a^{-6}-z^{8}-az^{7}+4z^{7}a^{-1}+3z^{7}a^{-3}-4z^{7}a^{-5}-2z^{7}a^{-7}+17z^{6}a^{-2}+18z^{6}a^{-4}+4z^{6}a^{-6}-z^{6}a^{-8}+4z^{6}+6az^{5}+3z^{5}a^{-1}+12z^{5}a^{-3}+25z^{5}a^{-5}+10z^{5}a^{-7}-18z^{4}a^{-2}-23z^{4}a^{-4}+4z^{4}a^{-8}+z^{4}-11az^{3}-19z^{3}a^{-1}-35z^{3}a^{-3}-39z^{3}a^{-5}-12z^{3}a^{-7}-10z^{2}a^{-2}-2z^{2}a^{-6}-2z^{2}a^{-8}-10z^{2}+10az+23za^{-1}+27za^{-3}+20za^{-5}+6za^{-7}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                 1 1 1   1 10               1 3 1     1 8             2 2 2       2 6           3 6 2         1 4         1 2 4           3 2       2 5 1             2 0     1 1 3               3 -2     2                   2 -4 1 1                     0 -6 1                       1

Integral Khovanov Homology (db, data source)

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

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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