L10a146

L10a145

L10a147

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Link Presentations

[edit Notes on L10a146's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w+2uv^{2}w^{2}-3uv^{2}w+uv^{2}+uvw^{3}-3uvw^{2}+2uvw-uw^{3}+uw^{2}-v^{3}w+v^{3}-2v^{2}w^{2}+3v^{2}w-v^{2}-vw^{3}+3vw^{2}-2vw-w^{2}}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{9}-2q^{8}+5q^{7}-7q^{6}+10q^{5}-9q^{4}+10q^{3}-8q^{2}+5q+q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}a^{-6}-2z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+z^{2}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-4z^{4}a^{-10}+4z^{2}a^{-10}+2z^{7}a^{-9}-6z^{5}a^{-9}+4z^{3}a^{-9}+3z^{8}a^{-8}-11z^{6}a^{-8}+18z^{4}a^{-8}-19z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+z^{9}a^{-7}+3z^{7}a^{-7}-15z^{5}a^{-7}+17z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+6z^{8}a^{-6}-18z^{6}a^{-6}+28z^{4}a^{-6}-30z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+z^{9}a^{-5}+5z^{7}a^{-5}-17z^{5}a^{-5}+21z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+3z^{8}a^{-4}-3z^{6}a^{-4}+3z^{4}a^{-4}-5z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+4z^{7}a^{-3}-6z^{5}a^{-3}+6z^{3}a^{-3}+3z^{6}a^{-2}-2z^{4}a^{-2}+2z^{5}a^{-1}-2z^{3}a^{-1}+z^{4}-2z^{2}+1$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 19                     1 1 17                   2 1 -1 15                 3     3 13               4 2     -2 11             6 3       3 9           4 5         1 7         6 5           1 5       2 4             2 3     3 6               -3 1   1 4                 3 -1   1                   -1 -3 1                     1

Integral Khovanov Homology (db, data source)

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

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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