L11a263

L11a262

L11a264

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a263 at Knotilus!

Link Presentations

[edit Notes on L11a263's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{3}v^{3}-3u^{3}v^{2}+u^{3}v-3u^{2}v^{3}+5u^{2}v^{2}-4u^{2}v+u^{2}+uv^{3}-4uv^{2}+5uv-3u+v^{2}-3v+2}{u^{3/2}v^{3/2}}}$ (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 q^{23/2}-3 q^{21/2}+5 q^{19/2}-8 q^{17/2}+11 q^{15/2}-12 q^{13/2}+11 q^{11/2}-10 q^{9/2}+7 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q}} (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-4z^{5}a^{-5}-4z^{5}a^{-7}+z^{5}a^{-9}+4z^{3}a^{-3}-4z^{3}a^{-5}-4z^{3}a^{-7}+3z^{3}a^{-9}+4za^{-3}-2za^{-5}-za^{-7}+za^{-9}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (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} +z^2 a^{-12} +4 z^7 a^{-11} -2 z^5 a^{-11} -3 z^3 a^{-11} +2 z a^{-11} +4 z^8 a^{-10} -5 z^6 a^{-10} +2 z^4 a^{-10} +z^2 a^{-10} +3 z^9 a^{-9} -5 z^7 a^{-9} +5 z^5 a^{-9} -5 z^3 a^{-9} +2 z a^{-9} +z^{10} a^{-8} +3 z^8 a^{-8} -13 z^6 a^{-8} +11 z^4 a^{-8} -3 z^2 a^{-8} +5 z^9 a^{-7} -15 z^7 a^{-7} +12 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +z^{10} a^{-6} +z^8 a^{-6} -12 z^6 a^{-6} +12 z^4 a^{-6} -3 z^2 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -3 z^5 a^{-5} +10 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} -z^2 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + a^{-3} 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   \ -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 3       -3 14             6 5         1 12           6 7           1 10         4 5             -1 8       3 6               3 6     2 4                 -2 4   1 4                   3 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

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

L11a262 L11a264