L10a145

L10a144

L10a146

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Link Presentations

[edit Notes on L10a145's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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} , 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 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{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3+t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2+t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)+t(1)-t(1) t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}} (db)
Jones 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 q^{11}-2 q^{10}+3 q^9-3 q^8+4 q^7-4 q^6+4 q^5-2 q^4+3 q^3-q^2+q} (db)
Signature	6 (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^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +5 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +7 z^2 a^{-8} +7 a^{-4} -11 a^{-6} +4 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} } (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 z^2 a^{-14} +2 z^3 a^{-13} +3 z^4 a^{-12} -3 z^2 a^{-12} + a^{-12} +3 z^5 a^{-11} -4 z^3 a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +3 z^7 a^{-9} -9 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -14 z^6 a^{-8} +20 z^4 a^{-8} -13 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} -2 z^7 a^{-7} -8 z^5 a^{-7} +19 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +4 z^8 a^{-6} -24 z^6 a^{-6} +46 z^4 a^{-6} -35 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} -5 z^7 a^{-5} +4 z^5 a^{-5} +9 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} -7 z^6 a^{-4} +17 z^4 a^{-4} -18 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} } (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums $\chi$ (fixed 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 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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   2 1 -1 19                 1     1 17               2 2     0 15             2 1       1 13           3 3         0 11         1 1           0 9       1 3             2 7     2 1               1 5   1 3                 2 3                       0 1 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=5} $i=7$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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=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 {\mathbb Z}^{2}\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=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}_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}^{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=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}} ${\mathbb {Z} }$

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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