L11n393

L11n392

L11n394

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Link Presentations

[edit Notes on L11n393's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X18,12,5,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {4, -1, 2, -5, -7, 8, 10, -4, -6, 9, 11, -2, 3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)^{3}+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-4q^{-5}+7q^{-4}+q^{3}-5q^{-3}-q^{2}+6q^{-2}-q-4q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)+a^{6}z^{-2}-a^{6}+2a^{4}z^{4}+5a^{4}z^{2}-2a^{4}z^{-2}+a^{4}-a^{2}z^{6}-4a^{2}z^{4}-4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}-z^{2}-1$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+4a^{7}z^{3}-a^{7}z+3a^{6}z^{8}-14a^{6}z^{6}+19a^{6}z^{4}-10a^{6}z^{2}+a^{6}z^{-2}+2a^{5}z^{9}-5a^{5}z^{7}-7a^{5}z^{5}+14a^{5}z^{3}-3a^{5}z-2a^{5}z^{-1}+8a^{4}z^{8}-39a^{4}z^{6}+55a^{4}z^{4}-31a^{4}z^{2}+2a^{4}z^{-2}+4a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-15a^{3}z^{5}+30a^{3}z^{3}-11a^{3}z-2a^{3}z^{-1}+5a^{2}z^{8}-27a^{2}z^{6}+z^{6}a^{-2}+41a^{2}z^{4}-5z^{4}a^{-2}-23a^{2}z^{2}+4z^{2}a^{-2}+a^{2}z^{-2}+5a^{2}-a^{-2}+3az^{7}+z^{7}a^{-1}-19az^{5}-7z^{5}a^{-1}+30az^{3}+10z^{3}a^{-1}-13az-4za^{-1}-z^{6}+2z^{2}+1$ (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 3 4 χ 7                       1 1 5                         0 3                 2 1 1   -2 1               3 1       2 -1             4 4 1       -1 -3           3 3 2         2 -5         3 4             1 -7       4 3 1             2 -9     2 5                 3 -11   1 2                   -1 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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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