L11n158

L11n157

L11n159

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Link Presentations

[edit Notes on L11n158's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X3,10,4,11 X2,15,3,16 X14,10,15,9 X11,19,12,18 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, -3, 11, 7, -8}, {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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