L11a293

L11a292

L11a294

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Link Presentations

[edit Notes on L11a293's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,14,21,13 X14,8,15,7 X16,6,17,5 X6,16,7,15 X4,21,5,22 X18,9,19,10 X22,17,9,18 X8,20,1,19
Gauss code	{1, -2, 3, -8, 6, -7, 5, -11}, {9, -1, 2, -3, 4, -5, 7, -6, 10, -9, 11, -4, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{4}-uv^{3}+uv^{2}-uv+u+v^{3}-v^{2}+v\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+5q^{3/2}-9{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {13}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z-a^{3}z^{7}-4a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z-az^{7}-4az^{5}+z^{5}a^{-1}-4az^{3}+3z^{3}a^{-1}+za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{3}+3a^{8}z^{4}+6a^{7}z^{5}-3a^{7}z^{3}+10a^{6}z^{6}-15a^{6}z^{4}+6a^{6}z^{2}+12a^{5}z^{7}-26a^{5}z^{5}+15a^{5}z^{3}-4a^{5}z+10a^{4}z^{8}-22a^{4}z^{6}+5a^{4}z^{4}+3a^{4}z^{2}+6a^{3}z^{9}-10a^{3}z^{7}-13a^{3}z^{5}+19a^{3}z^{3}-6a^{3}z+2a^{2}z^{10}+4a^{2}z^{8}+z^{8}a^{-2}-34a^{2}z^{6}-5z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-4z^{2}a^{-2}+9az^{9}+3z^{9}a^{-1}-38az^{7}-16z^{7}a^{-1}+47az^{5}+28z^{5}a^{-1}-17az^{3}-17z^{3}a^{-1}+2za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}-5z^{8}-7z^{6}+23z^{4}-11z^{2}-1$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     2   2 4                   3 1   -2 2                 6 2     4 0               5 3       -2 -2             9 6         3 -4           6 7           1 -6         6 7             -1 -8       4 6               2 -10     2 6                 -4 -12   1 4                   3 -14   2                     -2 -16 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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