L11a516

L11a515

L11a517

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a516 at Knotilus!

Link Presentations

[edit Notes on L11a516's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X18,9,19,10 X10,17,11,18 X20,15,21,16 X14,4,15,3 X4,22,5,21 X2738 X22,12,13,11 X12,14,7,13 X6,19,1,20
Gauss code	{1, -8, 6, -7, 2, -11}, {8, -1, 3, -4, 9, -10}, {10, -6, 5, -2, 4, -3, 11, -5, 7, -9}

A Braid Representative	{{{braid_table}}}
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} , 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} , ...)	$-{\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}t(2)^{2}+t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+t(2)^{2}+t(1)t(3)^{2}t(2)-t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)-t(1)t(3)^{2}-t(1)+2t(1)t(3)-t(3)\right)}{t(1)t(2)t(3)^{3/2}}}$ (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^4-4 q^3+9 q^2-15 q+21-22 q^{-1} +24 q^{-2} -19 q^{-3} +15 q^{-4} -9 q^{-5} +4 q^{-6} - q^{-7} } (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}+3z^{4}a^{4}+6z^{2}a^{4}+a^{4}z^{-2}+3a^{4}-2z^{6}a^{2}-6z^{4}a^{2}-7z^{2}a^{2}-2a^{2}z^{-2}-4a^{2}-z^{6}-z^{4}+z^{2}+z^{-2}+2+z^{4}a^{-2}+z^{2}a^{-2}$ (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 a^7 z^7-3 a^7 z^5+3 a^7 z^3-a^7 z+4 a^6 z^8-13 a^6 z^6+14 a^6 z^4-7 a^6 z^2+2 a^6+6 a^5 z^9-16 a^5 z^7+7 a^5 z^5+6 a^5 z^3-3 a^5 z+3 a^4 z^{10}+7 a^4 z^8-48 a^4 z^6+60 a^4 z^4+z^4 a^{-4} -29 a^4 z^2-a^4 z^{-2} +7 a^4+17 a^3 z^9-44 a^3 z^7+24 a^3 z^5+4 z^5 a^{-3} +3 a^3 z^3-z^3 a^{-3} -4 a^3 z+2 a^3 z^{-1} +3 a^2 z^{10}+19 a^2 z^8-73 a^2 z^6+9 z^6 a^{-2} +74 a^2 z^4-7 z^4 a^{-2} -33 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+11 a z^9-13 a z^7+14 z^7 a^{-1} -8 a z^5-18 z^5 a^{-1} +8 a z^3+7 z^3 a^{-1} -2 a z+2 a z^{-1} +16 z^8-29 z^6+20 z^4-9 z^2- z^{-2} +3} (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 $j$, alternation over $r$).

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 9                       1 1 7                     3   -3 5                   6 1   5 3                 9 3     -6 1               12 6       6 -1             12 11         -1 -3           12 10           2 -5         9 14             5 -7       6 10               -4 -9     3 9                 6 -11   1 6                   -5 -13   3                     3 -15 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}} $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-2$ 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}^{14}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ 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}^{6}} $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ 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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