L11a302

L11a301

L11a303

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a302 at Knotilus!

Link Presentations

[edit Notes on L11a302's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,6,17,5 X22,14,9,13 X20,16,21,15 X14,22,15,21 X18,8,19,7 X6,18,7,17 X8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -7, 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} , $w$, ...)	$-{\frac {2(u-1)(v-1)(uv+1)^{2}}{u^{3/2}v^{3/2}}}$ (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 -7 q^{9/2}+8 q^{7/2}-\frac{1}{q^{7/2}}-10 q^{5/2}+\frac{2}{q^{5/2}}+10 q^{3/2}-\frac{4}{q^{3/2}}+q^{15/2}-2 q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}}} (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^7 a^{-1} -z^7 a^{-3} +a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +4 a z^3-8 z^3 a^{-1} -8 z^3 a^{-3} +4 z^3 a^{-5} +4 a z-4 z a^{-1} -4 z a^{-3} +4 z a^{-5} +a z^{-1} - a^{-1} z^{-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 -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -4 z^9 a^{-3} -2 z^9 a^{-5} +z^8 a^{-2} +2 z^8 a^{-4} -2 z^8 a^{-6} -3 z^8-3 a z^7+4 z^7 a^{-1} +16 z^7 a^{-3} +7 z^7 a^{-5} -2 z^7 a^{-7} -2 a^2 z^6+2 z^6 a^{-2} -3 z^6 a^{-4} +5 z^6 a^{-6} -z^6 a^{-8} +9 z^6-a^3 z^5+9 a z^5+z^5 a^{-1} -30 z^5 a^{-3} -14 z^5 a^{-5} +7 z^5 a^{-7} +5 a^2 z^4-7 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} +4 z^4 a^{-8} -10 z^4+3 a^3 z^3-9 a z^3-13 z^3 a^{-1} +23 z^3 a^{-3} +18 z^3 a^{-5} -6 z^3 a^{-7} -a^2 z^2+2 z^2 a^{-4} +z^2 a^{-6} -4 z^2 a^{-8} +2 z^2-a^3 z+5 a z+6 z a^{-1} -6 z a^{-3} -5 z a^{-5} +z a^{-7} +1-a z^{-1} - a^{-1} z^{-1} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     1   1 12                   3 1   -2 10                 4 1     3 8               4 3       -1 6             6 4         2 4           4 4           0 2         5 6             -1 0       3 6               3 -2     1 3                 -2 -4   1 3                   2 -6   1                     -1 -8 1                       1

Integral Khovanov Homology (db, data source)

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

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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