L11a367

L11a366

L11a368

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Link Presentations

[edit Notes on L11a367's Link Presentations]

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

A Braid Representative
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{u^4 v^4-u^4 v^3-u^3 v^4+3 u^3 v^3-u^3 v^2-u^2 v^3+3 u^2 v^2-u^2 v-u v^2+3 u v-u-v+1}{u^2 v^2}} (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+11a^{7}z^{3}+7a^{7}z+a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-23a^{5}z^{5}-29a^{5}z^{3}-14a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+10a^{3}z^{3}+4a^{3}z$ (db)
Kauffman polynomial	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 a^{13} z^3-a^{13} z+2 a^{12} z^4-2 a^{12} z^2+2 a^{11} z^5-a^{11} z^3+2 a^{10} z^6-2 a^{10} z^4+a^{10} z^2+2 a^9 z^7-4 a^9 z^5+3 a^9 z^3-a^9 z+2 a^8 z^8-6 a^8 z^6+5 a^8 z^4-2 a^8 z^2+2 a^7 z^9-9 a^7 z^7+15 a^7 z^5-16 a^7 z^3+8 a^7 z-a^7 z^{-1} +a^6 z^{10}-3 a^6 z^8-2 a^6 z^6+9 a^6 z^4-7 a^6 z^2+a^6+4 a^5 z^9-24 a^5 z^7+48 a^5 z^5-42 a^5 z^3+15 a^5 z-a^5 z^{-1} +a^4 z^{10}-4 a^4 z^8+10 a^4 z^4-6 a^4 z^2+2 a^3 z^9-13 a^3 z^7+27 a^3 z^5-21 a^3 z^3+5 a^3 z+a^2 z^8-6 a^2 z^6+10 a^2 z^4-4 a^2 z^2} (db)

Khovanov Homology

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

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     1   1 -2                   1 1   0 -4                 3 1     2 -6               2 2       0 -8             3 2         1 -10           2 2           0 -12         3 3             0 -14       1 2               1 -16     2 3                 -1 -18   1 2                   1 -20   1                     -1 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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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