L11n349

L11n348

L11n350

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Link Presentations

[edit Notes on L11n349's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,18,13,17 X9,17,10,16 X15,11,16,10 X12,20,5,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -6, 7, 3, -8}, {11, -2, -7, 6, 5, -4, 8, -3, 9, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	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 \frac{u v^2 w^3-2 u v^2 w^2+2 u v^2 w-u v^2-u v w^3+2 u v w^2-3 u v w+2 u v+u w+v^3 \left(-w^2\right)-2 v^2 w^3+3 v^2 w^2-2 v^2 w+v^2+v w^3-2 v w^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}}} (db)
Jones polynomial	$-q^{6}+5q^{5}-7q^{4}+10q^{3}+q^{-3}-10q^{2}-2q^{-2}+10q+6q^{-1}-8$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+2z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}-4z^{2}+2a^{2}-5a^{-2}+3a^{-4}-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+5z^{4}a^{-6}-z^{2}a^{-6}-2a^{-6}+2z^{7}a^{-5}+z^{3}a^{-5}+za^{-5}+3z^{8}a^{-4}-5z^{6}a^{-4}+8z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+3a^{-4}+z^{9}a^{-3}+5z^{7}a^{-3}-15z^{5}a^{-3}+15z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+6z^{8}a^{-2}+a^{2}z^{6}-14z^{6}a^{-2}-4a^{2}z^{4}+14z^{4}a^{-2}+5a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-2a^{2}+9a^{-2}+z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-5az^{5}-20z^{5}a^{-1}+2az^{3}+17z^{3}a^{-1}+az-8za^{-1}+2a^{-1}z^{-1}+3z^{8}-8z^{6}+7z^{4}-6z^{2}-z^{-2}+3$ (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 χ 13                   1 -1 11                 4   4 9               5 3   -2 7             5 2     3 5           5 5       0 3         5 5         0 1       4 6           2 -1     2 4             -2 -3     4               4 -5 1 2                 -1 -7 1                   1

Integral Khovanov Homology (db, data source)

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

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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