L10a68

L10a67

L10a69

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Link Presentations

[edit Notes on L10a68's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,18,7,17 X18,12,19,11 X12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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