L10a57

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Link Presentations

[edit Notes on L10a57's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X14,6,15,5 X16,11,17,12 X20,17,7,18 X18,14,19,13 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 4, -7, 6, -3, 8, -4, 5, -6, 7, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{4}+t(2)^{4}-t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-9t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+5t(1)t(2)-t(2)+t(1)^{2}-t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {12}{q^{5/2}}}-6q^{3/2}+{\frac {12}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+za^{5}+a^{5}z^{-1}-z^{5}a^{3}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{5}a+2za+2az^{-1}+2z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}-za^{-3}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+3a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{7}-12a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+4a^{4}z^{8}-6a^{4}z^{6}+3a^{4}z^{4}+a^{3}z^{9}+9a^{3}z^{7}-27a^{3}z^{5}+z^{5}a^{-3}+31a^{3}z^{3}-2z^{3}a^{-3}-14a^{3}z+za^{-3}+2a^{3}z^{-1}+7a^{2}z^{8}-12a^{2}z^{6}+3z^{6}a^{-2}+9a^{2}z^{4}-6z^{4}a^{-2}-3a^{2}z^{2}+3z^{2}a^{-2}+a^{2}+az^{9}+8az^{7}+4z^{7}a^{-1}-21az^{5}-6z^{5}a^{-1}+22az^{3}+2z^{3}a^{-1}-12az-2za^{-1}+2az^{-1}+a^{-1}z^{-1}+3z^{8}-6z^{4}+2z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   2   -2 4                 4 1   3 2               5 2     -3 0             7 4       3 -2           6 6         0 -4         6 6           0 -6       4 7             3 -8     2 5               -3 -10   1 4                 3 -12   2                   -2 -14 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ 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}^{2}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ 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}_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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