L11a455

L11a454

L11a456

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Link Presentations

[edit Notes on L11a455's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X16,10,17,9 X8,18,9,17 X22,16,13,15 X12,20,5,19 X10,22,11,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 5, -9, 3, -8}, {11, -2, 7, -5, 6, -4, 8, -3, 9, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{3}w-uv^{3}+2uv^{2}w^{2}-4uv^{2}w+2uv^{2}+uvw^{3}-4uvw^{2}+3uvw-uw^{3}+2uw^{2}-2v^{3}w+v^{3}-3v^{2}w^{2}+4v^{2}w-v^{2}-2vw^{3}+4vw^{2}-2vw+w^{3}-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-5q^{5}+10q^{4}+q^{-4}-11q^{3}-2q^{-3}+13q^{2}+5q^{-2}-13q-8q^{-1}+12$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}-2z^{2}a^{2}+z^{4}-2z^{2}+z^{-2}+2z^{4}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{4}a^{-4}+a^{-4}z^{-2}+2a^{-4}-z^{2}a^{-6}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+3z^{9}a^{-1}+6z^{9}a^{-3}+3z^{9}a^{-5}+7z^{8}a^{-2}+5z^{8}a^{-4}+3z^{8}a^{-6}+5z^{8}+4az^{7}-14z^{7}a^{-3}-9z^{7}a^{-5}+z^{7}a^{-7}+3a^{2}z^{6}-30z^{6}a^{-2}-29z^{6}a^{-4}-13z^{6}a^{-6}-11z^{6}+2a^{3}z^{5}-4az^{5}-16z^{5}a^{-1}-2z^{5}a^{-3}+4z^{5}a^{-5}-4z^{5}a^{-7}+a^{4}z^{4}-2a^{2}z^{4}+39z^{4}a^{-2}+39z^{4}a^{-4}+18z^{4}a^{-6}+15z^{4}-2a^{3}z^{3}+2az^{3}+23z^{3}a^{-1}+17z^{3}a^{-3}+2z^{3}a^{-5}+4z^{3}a^{-7}-2a^{4}z^{2}-29z^{2}a^{-2}-26z^{2}a^{-4}-9z^{2}a^{-6}-10z^{2}-11za^{-1}-11za^{-3}+a^{4}+13a^{-2}+8a^{-4}+5+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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 15                       1 -1 13                     2   2 11                   3 1   -2 9                 7 2     5 7               6 5       -1 5             7 5         2 3           6 6           0 1         6 7             -1 -1       3 7               4 -3     2 5                 -3 -5     3                   3 -7 1 2                     -1 -9 1                       1

Integral Khovanov Homology (db, data source)

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

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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