L11a317

L11a316

L11a318

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Link Presentations

[edit Notes on L11a317's Link Presentations]

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

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} , $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{3}-2u^{2}v^{2}+u^{2}v+uv^{4}-3uv^{3}+6uv^{2}-3uv+u+v^{3}-2v^{2}+v\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {17}{q^{9/2}}}-q^{7/2}+{\frac {24}{q^{7/2}}}+5q^{5/2}-{\frac {29}{q^{5/2}}}-12q^{3/2}+{\frac {28}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {10}{q^{11/2}}}+19{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+5a^{3}z^{3}-z^{3}a^{-1}-2a^{5}z+5a^{3}z-3az-a^{5}z^{-1}+3a^{3}z^{-1}-2az^{-1}$ (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 -4 a^4 z^{10}-4 a^2 z^{10}-9 a^5 z^9-21 a^3 z^9-12 a z^9-8 a^6 z^8-11 a^4 z^8-19 a^2 z^8-16 z^8-4 a^7 z^7+16 a^5 z^7+40 a^3 z^7+8 a z^7-12 z^7 a^{-1} -a^8 z^6+17 a^6 z^6+38 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5-11 a^5 z^5-26 a^3 z^5+9 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-13 a^6 z^4-32 a^4 z^4-29 a^2 z^4+3 z^4 a^{-2} -9 z^4-4 a^7 z^3+7 a^5 z^3+15 a^3 z^3-a z^3-5 z^3 a^{-1} -a^8 z^2+5 a^6 z^2+12 a^4 z^2+8 a^2 z^2+2 z^2-4 a^5 z-9 a^3 z-5 a z-a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     4   -4 4                   8 1   7 2                 11 4     -7 0               15 8       7 -2             15 13         -2 -4           14 13           1 -6         10 15             5 -8       7 14               -7 -10     3 10                 7 -12   1 7                   -6 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ 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 r=-7} ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ 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}} 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} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ 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=-4} ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ 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}^{7}} $r=-3$ 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}^{14}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=-2$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$ 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}^{15}} 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=0} ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ 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}^{4}\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=3} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=4$ 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}_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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