L11a336

L11a335

L11a337

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a336 at Knotilus!

Link Presentations

[edit Notes on L11a336's Link Presentations]

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

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} , ...)	$-{\frac {3u^{3}v^{2}-3u^{3}v+u^{3}+4u^{2}v^{3}-10u^{2}v^{2}+11u^{2}v-4u^{2}-4uv^{3}+11uv^{2}-10uv+4u+v^{3}-3v^{2}+3v}{u^{3/2}v^{3/2}}}$ (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 -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{20}{q^{13/2}}+\frac{23}{q^{15/2}}-\frac{23}{q^{17/2}}+\frac{21}{q^{19/2}}-\frac{16}{q^{21/2}}+\frac{10}{q^{23/2}}-\frac{5}{q^{25/2}}+\frac{1}{q^{27/2}}} (db)
Signature	-5 (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^{13}+4 z^3 a^{11}+5 z a^{11}-3 z^5 a^9-6 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^5 a^7-7 z^3 a^7-5 z a^7-a^7 z^{-1} -z^5 a^5-2 z^3 a^5-z a^5} (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^{16} z^6-a^{16} z^4+5 a^{15} z^7-10 a^{15} z^5+4 a^{15} z^3+a^{15} z+9 a^{14} z^8-20 a^{14} z^6+11 a^{14} z^4-2 a^{14} z^2+7 a^{13} z^9-5 a^{13} z^7-17 a^{13} z^5+14 a^{13} z^3-2 a^{13} z+2 a^{12} z^{10}+17 a^{12} z^8-51 a^{12} z^6+43 a^{12} z^4-15 a^{12} z^2+13 a^{11} z^9-18 a^{11} z^7-3 a^{11} z^5+7 a^{11} z^3+a^{11} z+2 a^{10} z^{10}+15 a^{10} z^8-40 a^{10} z^6+36 a^{10} z^4-11 a^{10} z^2+6 a^9 z^9-2 a^9 z^7-7 a^9 z^5+9 a^9 z^3-3 a^9 z+a^9 z^{-1} +7 a^8 z^8-7 a^8 z^6+a^8 z^4+3 a^8 z^2-a^8+6 a^7 z^7-10 a^7 z^5+10 a^7 z^3-6 a^7 z+a^7 z^{-1} +3 a^6 z^6-4 a^6 z^4+a^6 z^2+a^5 z^5-2 a^5 z^3+a^5 z} (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 8 3     -5 -12               12 5       7 -14             11 8         -3 -16           12 12           0 -18         9 11             2 -20       7 12               -5 -22     4 10                 6 -24   1 6                   -5 -26   4                     4 -28 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}} $i=-6$ 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 i=-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=-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}} 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=-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}^{4}\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=-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}^{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}^{4}} $r=-8$ 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}^{10}\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}^{7}} $r=-7$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-6$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-5$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ 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}_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=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}} 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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