L11a362

L11a361

L11a363

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a362 at Knotilus!

Link Presentations

[edit Notes on L11a362's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-4t(2)^{3}t(1)^{3}+3t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+3t(2)^{3}t(1)^{2}-5t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+3t(2)^{2}t(1)-4t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+5q^{19/2}-8q^{17/2}+10q^{15/2}-11q^{13/2}+10q^{11/2}-9q^{9/2}+6q^{7/2}-4q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (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^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -3 z^3 a^{-5} -4 z^3 a^{-7} +3 z^3 a^{-9} +3 z a^{-3} +z a^{-5} -2 z a^{-7} +z a^{-9} + a^{-5} z^{-1} - a^{-7} 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^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -4 z^3 a^{-13} +z a^{-13} +4 z^6 a^{-12} -4 z^4 a^{-12} +4 z^7 a^{-11} -3 z^5 a^{-11} -2 z^3 a^{-11} +2 z a^{-11} +4 z^8 a^{-10} -7 z^6 a^{-10} +6 z^4 a^{-10} -z^2 a^{-10} +3 z^9 a^{-9} -7 z^7 a^{-9} +9 z^5 a^{-9} -6 z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +2 z^8 a^{-8} -13 z^6 a^{-8} +16 z^4 a^{-8} -6 z^2 a^{-8} +5 z^9 a^{-7} -19 z^7 a^{-7} +25 z^5 a^{-7} -17 z^3 a^{-7} +6 z a^{-7} - a^{-7} z^{-1} +z^{10} a^{-6} -11 z^6 a^{-6} +16 z^4 a^{-6} -8 z^2 a^{-6} + a^{-6} +2 z^9 a^{-5} -7 z^7 a^{-5} +5 z^5 a^{-5} -2 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -9 z^6 a^{-4} +11 z^4 a^{-4} -4 z^2 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +7 z^3 a^{-3} -3 z a^{-3} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ 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 9 χ 24                       1 -1 22                     2   2 20                   3 1   -2 18                 5 2     3 16               6 4       -2 14             5 4         1 12           5 6           1 10         4 5             -1 8       2 5               3 6     2 4                 -2 4   1 3                   2 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=4$ $i=6$ $r=-2$ 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}} $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ 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}^{4}\oplus{\mathbb Z}_2^{2}} ${\mathbb {Z} }^{2}$ $r=2$ 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}^{5}\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}} 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} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ 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}^{6}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=5$ 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}^{4}\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}} $r=6$ 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}^{4}\oplus{\mathbb Z}_2^{4}} ${\mathbb {Z} }^{5}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ 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=9} ${\mathbb {Z} }_{2}$ 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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