L11a235

L11a234

L11a236

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a235 at Knotilus!

Link Presentations

[edit Notes on L11a235's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,13,21,14 X16,9,17,10 X10,19,11,20 X22,15,7,16 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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} , $w$, ...)	${\frac {-t(2)^{6}-3t(1)t(2)^{5}+3t(2)^{5}-3t(1)^{2}t(2)^{4}+6t(1)t(2)^{4}-4t(2)^{4}+4t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+4t(2)^{3}-4t(1)^{2}t(2)^{2}+6t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {16}{q^{13/2}}}+{\frac {17}{q^{15/2}}}-{\frac {18}{q^{17/2}}}+{\frac {15}{q^{19/2}}}-{\frac {11}{q^{21/2}}}+{\frac {7}{q^{23/2}}}-{\frac {3}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+4a^{11}z^{3}+10a^{11}z+5a^{11}z^{-1}-3a^{9}z^{5}-9a^{9}z^{3}-8a^{9}z-3a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-4a^{7}z-a^{5}z^{5}-2a^{5}z^{3}$ (db)
Kauffman polynomial	$a^{16}z^{6}-3a^{16}z^{4}+3a^{16}z^{2}-a^{16}+3a^{15}z^{7}-8a^{15}z^{5}+5a^{15}z^{3}+4a^{14}z^{8}-7a^{14}z^{6}-2a^{14}z^{4}+4a^{14}z^{2}+4a^{13}z^{9}-7a^{13}z^{7}+5a^{13}z^{5}-10a^{13}z^{3}+7a^{13}z-2a^{13}z^{-1}+2a^{12}z^{10}+2a^{12}z^{8}-12a^{12}z^{6}+17a^{12}z^{4}-17a^{12}z^{2}+5a^{12}+10a^{11}z^{9}-31a^{11}z^{7}+51a^{11}z^{5}-47a^{11}z^{3}+22a^{11}z-5a^{11}z^{-1}+2a^{10}z^{10}+5a^{10}z^{8}-23a^{10}z^{6}+36a^{10}z^{4}-21a^{10}z^{2}+5a^{10}+6a^{9}z^{9}-15a^{9}z^{7}+23a^{9}z^{5}-18a^{9}z^{3}+11a^{9}z-3a^{9}z^{-1}+7a^{8}z^{8}-16a^{8}z^{6}+15a^{8}z^{4}-3a^{8}z^{2}+6a^{7}z^{7}-14a^{7}z^{5}+12a^{7}z^{3}-4a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+a^{5}z^{5}-2a^{5}z^{3}$ (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 $r$).

\   r    \    j   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 7 3     -4 -12               9 4       5 -14             8 7         -1 -16           10 9           1 -18         6 9             3 -20       5 9               -4 -22     2 6                 4 -24   1 5                   -4 -26   2                     2 -28 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-8$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-7$ 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}^{9}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=-6$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ 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}^{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 r=-5} ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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 r=-4} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ 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}} $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ 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}} $r=0$ ${\mathbb {Z} }$ 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.

L11a234 L11a236