L10a151

L10a150

L10a152

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Link Presentations

[edit Notes on L10a151's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-3t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+3q^{4}+3q^{-4}-6q^{3}-6q^{-3}+11q^{2}+11q^{-2}-12q-12q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-6z^{2}-a^{4}+4a^{2}+4a^{-2}-a^{-4}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+9az^{7}+9z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+4a^{2}z^{6}+4z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}+a^{5}z^{5}-4a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-18a^{2}z^{4}-18z^{4}a^{-2}-6z^{4}a^{-4}-24z^{4}-2a^{5}z^{3}-2a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-5}+5a^{4}z^{2}+18a^{2}z^{2}+18z^{2}a^{-2}+5z^{2}a^{-4}+26z^{2}+a^{5}z+3a^{3}z+4az+4za^{-1}+3za^{-3}+za^{-5}-2a^{4}-8a^{2}-8a^{-2}-2a^{-4}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   2   2 7                 4 1   -3 5               7 2     5 3             5 4       -1 1           9 7         2 -1         7 9           2 -3       4 5             -1 -5     2 7               5 -7   1 4                 -3 -9   2                   2 -11 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}} ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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