L10a12

L10a11

L10a13

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Link Presentations

[edit Notes on L10a12's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X18,11,19,12 X20,16,5,15 X16,20,17,19 X12,17,13,18 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 8, -3, 5, -6, 7, -4, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}-6uv^{2}+9uv-3u-3v^{3}+9v^{2}-6v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-7q^{3/2}+{\frac {13}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+10{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+a^{3}z-a^{3}z^{-1}-za^{-3}-az^{5}+2z^{3}a^{-1}+az+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-3a^{4}z^{8}-7a^{2}z^{8}-4z^{8}-3a^{5}z^{7}-8a^{3}z^{7}-10az^{7}-5z^{7}a^{-1}-2a^{6}z^{6}+2a^{4}z^{6}+10a^{2}z^{6}-3z^{6}a^{-2}+3z^{6}-a^{7}z^{5}+3a^{5}z^{5}+20a^{3}z^{5}+26az^{5}+9z^{5}a^{-1}-z^{5}a^{-3}+3a^{6}z^{4}-a^{4}z^{4}-7a^{2}z^{4}+5z^{4}a^{-2}+2z^{4}+3a^{7}z^{3}+a^{5}z^{3}-20a^{3}z^{3}-26az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}+3a^{4}z^{2}+6a^{2}z^{2}-2z^{2}a^{-2}+z^{2}-3a^{7}z-2a^{5}z+9a^{3}z+13az+4za^{-1}-za^{-3}-a^{6}-2a^{4}-3a^{2}-1+a^{7}z^{-1}+a^{5}z^{-1}-a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   2   -2 4                 5 1   4 2               5 2     -3 0             7 5       2 -2           7 6         -1 -4         5 6           -1 -6       4 7             3 -8     2 5               -3 -10   1 5                 4 -12   1                   -1 -14 1                     1

Integral Khovanov Homology (db, data source)

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

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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