L11a454

L11a453

L11a455

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Link Presentations

[edit Notes on L11a454's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,18,13,17 X16,9,17,10 X10,15,11,16 X12,20,5,19 X8,22,9,21 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 6, -7, 3, -8}, {11, -2, 7, -6, 5, -4, 8, -3, 9, -5}

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} , 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(3)^2 t(2)^3+t(1) t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+7 t(3)^2 t(2)^2-3 t(1) t(2)^2+6 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+3 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+2 t(1) t(2)-7 t(1) t(3) t(2)+4 t(3) t(2)-t(2)+t(1) t(3)^3-t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}} (db)
Jones polynomial	$q^{-6}-q^{5}-3q^{-5}+4q^{4}+8q^{-4}-10q^{3}-14q^{-3}+17q^{2}+21q^{-2}-21q-23q^{-1}+25$ (db)
Signature	0 (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 a^6-3 a^4 z^2+a^4 z^{-2} -z^2 a^{-4} -a^4+3 a^2 z^4+2 z^4 a^{-2} +a^2 z^2-2 a^2 z^{-2} -3 a^2- a^{-2} -z^6+2 z^2+ z^{-2} +4} (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 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+13 a z^9+8 z^9 a^{-1} +5 a^4 z^8+13 a^2 z^8+12 z^8 a^{-2} +20 z^8+3 a^5 z^7-3 a^3 z^7-16 a z^7-z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-9 a^4 z^6-39 a^2 z^6-20 z^6 a^{-2} +4 z^6 a^{-4} -53 z^6-7 a^5 z^5-10 a^3 z^5-8 a z^5-19 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+5 a^4 z^4+41 a^2 z^4+16 z^4 a^{-2} -4 z^4 a^{-4} +53 z^4+5 a^5 z^3+14 a^3 z^3+19 a z^3+19 z^3 a^{-1} +8 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2-3 a^4 z^2-27 a^2 z^2-10 z^2 a^{-2} +z^2 a^{-4} -32 z^2-a^5 z-8 a^3 z-12 a z-7 z a^{-1} -2 z a^{-3} -a^6+3 a^4+11 a^2+3 a^{-2} +11+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} } (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 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 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   7 1   -6 5                 10 3     7 3               11 7       -4 1             14 10         4 -1           12 14           2 -3         9 11             -2 -5       5 12               7 -7     3 9                 -6 -9   1 6                   5 -11   2                     -2 -13 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=-1} $i=1$ 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 r=-6} ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ 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}^{9}} 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}^{11}\oplus{\mathbb Z}_2^{12}} 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}^{12}} 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} 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^{11}} 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}} 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} ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ 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}^{11}} 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}^{7}\oplus{\mathbb Z}_2^{10}} 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}^{10}} 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} 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}\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}} 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} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ 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}} 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} 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.

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