L11a277

L11a276

L11a278

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Link Presentations

[edit Notes on L11a277's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X8,9,1,10 X4,13,5,14 X20,15,21,16 X18,8,19,7 X6,20,7,19 X22,17,9,18 X16,21,17,22
Gauss code	{1, -2, 3, -6, 4, -9, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -11, 10, -8, 9, -7, 11, -10}

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} , 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} , $w$, ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-t(1)^{3}t(2)^{4}+t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(1)^{3}t(2)^{3}-t(1)^{2}t(2)^{3}+t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+t(1)t(2)-t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {3}{q^{9/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {3}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{7}\right)-6a^{9}z^{5}-11a^{9}z^{3}-7a^{9}z-a^{9}z^{-1}+a^{7}z^{9}+8a^{7}z^{7}+23a^{7}z^{5}+30a^{7}z^{3}+18a^{7}z+3a^{7}z^{-1}-a^{5}z^{7}-7a^{5}z^{5}-16a^{5}z^{3}-13a^{5}z-2a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}-2z^{3}a^{15}-3z^{4}a^{14}+2z^{2}a^{14}-4z^{5}a^{13}+7z^{3}a^{13}-2za^{13}-4z^{6}a^{12}+9z^{4}a^{12}-2z^{2}a^{12}-4z^{7}a^{11}+12z^{5}a^{11}-5z^{3}a^{11}-za^{11}-4z^{8}a^{10}+17z^{6}a^{10}-19z^{4}a^{10}+7z^{2}a^{10}-a^{10}-3z^{9}a^{9}+15z^{7}a^{9}-22z^{5}a^{9}+13z^{3}a^{9}-6za^{9}+a^{9}z^{-1}-z^{10}a^{8}+2z^{8}a^{8}+12z^{6}a^{8}-34z^{4}a^{8}+23z^{2}a^{8}-3a^{8}-4z^{9}a^{7}+27z^{7}a^{7}-61z^{5}a^{7}+56z^{3}a^{7}-22za^{7}+3a^{7}z^{-1}-z^{10}a^{6}+6z^{8}a^{6}-9z^{6}a^{6}-3z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-z^{9}a^{5}+8z^{7}a^{5}-23z^{5}a^{5}+29z^{3}a^{5}-15za^{5}+2a^{5}z^{-1}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -2                       1 1 -4                         0 -6                   3 1   2 -8                 1 1     0 -10               3 2       1 -12             2 1         -1 -14           3 3           0 -16         2 3             1 -18       2 2               0 -20     1 2                 1 -22   1 2                   -1 -24   1                     1 -26 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-8$ $i=-6$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ 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}} 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}} 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} 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}} 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} 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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