L10a156

L10a155

L10a157

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Link Presentations

[edit Notes on L10a156's Link Presentations]

Planar diagram presentation	X8192 X18,10,19,9 X6,18,1,17 X16,7,17,8 X10,4,11,3 X14,6,15,5 X4,14,5,13 X20,11,13,12 X12,15,7,16 X2,19,3,20
Gauss code	{1, -10, 5, -7, 6, -3}, {4, -1, 2, -5, 8, -9}, {7, -6, 9, -4, 3, -2, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, $w$, ...)	${\frac {(t(3)-1)(-t(2)t(1)+t(2)t(3)t(1)+t(1)+t(2)-1)(-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+t(3)-1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+4q^{4}+4q^{-4}-8q^{3}-8q^{-3}+13q^{2}+13q^{-2}-15q-15q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-3z^{4}a^{-2}+8z^{4}-2a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}+z^{5}a^{-5}-a^{5}z^{3}-z^{3}a^{-5}+4a^{4}z^{6}+4z^{6}a^{-4}-6a^{4}z^{4}-6z^{4}a^{-4}+2a^{4}z^{2}+2z^{2}a^{-4}+7a^{3}z^{7}+7z^{7}a^{-3}-11a^{3}z^{5}-11z^{5}a^{-3}+4a^{3}z^{3}+4z^{3}a^{-3}+7a^{2}z^{8}+7z^{8}a^{-2}-9a^{2}z^{6}-9z^{6}a^{-2}+2a^{2}z^{4}+2z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+3az^{9}+3z^{9}a^{-1}+7az^{7}+7z^{7}a^{-1}-19az^{5}-19z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+14z^{8}-26z^{6}+16z^{4}-2z^{2}+2z^{-2}-3$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over 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} ).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   3   3 7                 5 1   -4 5               8 3     5 3             8 6       -2 1           10 7         3 -1         7 10           3 -3       6 8             -2 -5     3 8               5 -7   1 5                 -4 -9   3                   3 -11 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=1} $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=-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}^{5}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ 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}^{3}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L10a155 L10a157