L11a394

L11a393

L11a395

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Link Presentations

[edit Notes on L11a394's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X22,16,9,15 X18,12,19,11 X14,20,15,19 X20,14,21,13 X12,22,13,21 X8,18,5,17 X16,8,17,7 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, 4, -7, 6, -5, 3, -9, 8, -4, 5, -6, 7, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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 v} , $w$, ...)	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{-2 t(1) t(3)^3+2 t(1) t(2) t(3)^3-4 t(2) t(3)^3+2 t(3)^3+5 t(1) t(3)^2-3 t(1) t(2) t(3)^2+6 t(2) t(3)^2-3 t(3)^2-6 t(1) t(3)+3 t(1) t(2) t(3)-5 t(2) t(3)+3 t(3)+4 t(1)-2 t(1) t(2)+2 t(2)-2}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}} (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+11q^{5}-16q^{4}+18q^{3}+q^{-3}-16q^{2}-2q^{-2}+16q+7q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+3z^{4}a^{-4}+6z^{2}a^{-4}+3a^{-4}z^{-2}+6a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}-2a^{-2}z^{-2}+2a^{2}-3a^{-2}-2z^{4}-4z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+6z^{7}a^{-7}-14z^{5}a^{-7}+14z^{3}a^{-7}-7za^{-7}+2a^{-7}z^{-1}+7z^{8}a^{-6}-16z^{6}a^{-6}+17z^{4}a^{-6}-5z^{2}a^{-6}-a^{-6}z^{-2}+a^{-6}+4z^{9}a^{-5}+z^{7}a^{-5}-21z^{5}a^{-5}+40z^{3}a^{-5}-27za^{-5}+8a^{-5}z^{-1}+z^{10}a^{-4}+10z^{8}a^{-4}-25z^{6}a^{-4}+23z^{4}a^{-4}-9z^{2}a^{-4}-3a^{-4}z^{-2}+5a^{-4}+6z^{9}a^{-3}-2z^{7}a^{-3}-23z^{5}a^{-3}+43z^{3}a^{-3}-34za^{-3}+10a^{-3}z^{-1}+z^{10}a^{-2}+6z^{8}a^{-2}+a^{2}z^{6}-11z^{6}a^{-2}-4a^{2}z^{4}+2z^{4}a^{-2}+6a^{2}z^{2}-4z^{2}a^{-2}+a^{2}z^{-2}-2a^{-2}z^{-2}-4a^{2}+4a^{-2}+2z^{9}a^{-1}+2az^{7}+5z^{7}a^{-1}-4az^{5}-21z^{5}a^{-1}+19z^{3}a^{-1}+4az-10za^{-1}-2az^{-1}+2a^{-1}z^{-1}+3z^{8}-4z^{6}-3z^{4}+6z^{2}+z^{-2}-3$ (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 6 7 χ 17                       1 -1 15                     2   2 13                   5 1   -4 11                 6 2     4 9               10 5       -5 7             8 6         2 5           8 10           2 3         8 8             0 1       5 11               6 -1     2 5                 -3 -3     5                   5 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ 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 i=3} $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{2}$ 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 r=-2} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ 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}^{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 r=-1} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ 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}^{8}} $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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 r=2} ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ 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}^{6}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=4$ 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}^{5}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=5$ 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^{5}} ${\mathbb {Z} }^{5}$ $r=6$ 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}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ 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 r=7} ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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