L10a36

L10a35

L10a37

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Link Presentations

[edit Notes on L10a36's Link Presentations]

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

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} , $v$, $w$, ...)	$-{\frac {(u-1)(v-2)(v-1)(2v-1)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {13}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT 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 -z a^7-a^7 z^{-1} +2 z^3 a^5+3 z a^5+2 a^5 z^{-1} -z^5 a^3-z^3 a^3-z a^3-z^5 a-z^3 a-z a-a z^{-1} +z^3 a^{-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 a^8 z^6-4 a^8 z^4+5 a^8 z^2-2 a^8+2 a^7 z^7-6 a^7 z^5+5 a^7 z^3-2 a^7 z+a^7 z^{-1} +2 a^6 z^8-2 a^6 z^6-7 a^6 z^4+11 a^6 z^2-5 a^6+a^5 z^9+4 a^5 z^7-15 a^5 z^5+13 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +6 a^4 z^8-9 a^4 z^6-2 a^4 z^4+7 a^4 z^2-3 a^4+a^3 z^9+9 a^3 z^7-21 a^3 z^5+13 a^3 z^3-2 a^3 z+4 a^2 z^8+a^2 z^6-8 a^2 z^4+z^4 a^{-2} +2 a^2 z^2+a^2+7 a z^7-8 a z^5+4 z^5 a^{-1} +2 a z^3-3 z^3 a^{-1} +a z-a z^{-1} +7 z^6-8 z^4+z^2} (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                     1 -1 4                   3   3 2                 4 1   -3 0               6 3     3 -2             7 6       -1 -4           6 4         2 -6         4 7           3 -8       4 6             -2 -10     1 4               3 -12   1 4                 -3 -14   1                   1 -16 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

L10a35 L10a37