L11a360

L11a359

L11a361

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a360 at Knotilus!

Link Presentations

[edit Notes on L11a360's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{2}+uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$2q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}+q^{25/2}-2q^{23/2}+2q^{21/2}-3q^{19/2}+3q^{17/2}-3q^{15/2}+3q^{13/2}-3q^{11/2}$ (db)
Signature	7 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-7}+z^{7}a^{-5}-8z^{7}a^{-7}+z^{7}a^{-9}+7z^{5}a^{-5}-22z^{5}a^{-7}+6z^{5}a^{-9}+15z^{3}a^{-5}-25z^{3}a^{-7}+10z^{3}a^{-9}+10za^{-5}-11za^{-7}+4za^{-9}+a^{-5}z^{-1}-a^{-7}z^{-1}$ (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^{10} a^{-6} -z^{10} a^{-8} -z^9 a^{-5} -3 z^9 a^{-7} -2 z^9 a^{-9} +7 z^8 a^{-6} +5 z^8 a^{-8} -2 z^8 a^{-10} +8 z^7 a^{-5} +21 z^7 a^{-7} +11 z^7 a^{-9} -2 z^7 a^{-11} -15 z^6 a^{-6} -5 z^6 a^{-8} +8 z^6 a^{-10} -2 z^6 a^{-12} -22 z^5 a^{-5} -48 z^5 a^{-7} -18 z^5 a^{-9} +6 z^5 a^{-11} -2 z^5 a^{-13} +9 z^4 a^{-6} -3 z^4 a^{-8} -6 z^4 a^{-10} +4 z^4 a^{-12} -2 z^4 a^{-14} +25 z^3 a^{-5} +42 z^3 a^{-7} +11 z^3 a^{-9} -2 z^3 a^{-11} +2 z^3 a^{-13} -2 z^3 a^{-15} +2 z^2 a^{-6} +3 z^2 a^{-8} -z^2 a^{-10} +z^2 a^{-14} -z^2 a^{-16} -11 z a^{-5} -13 z a^{-7} -3 z a^{-9} -z a^{-11} +z a^{-13} +z a^{-15} - a^{-6} + a^{-5} z^{-1} + a^{-7} z^{-1} } (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 $r$).

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 26                       1 -1 24                     1   1 22                   1 1   0 20                 2 1     1 18               2 2       0 16             1 1         0 14           2 2           0 12         1 1             0 10       1 2               1 8     1 1                 0 6   1 2                   1 4                         0 2 1                       1

Integral Khovanov Homology (db, data source)

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

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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