L10a136

L10a135

L10a137

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a136 at Knotilus!

Link Presentations

[edit Notes on L10a136's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,7,11,8 X8,19,9,20 X18,12,19,11 X16,13,17,14 X14,6,15,5 X10,16,5,15 X2,9,3,10 X4,18,1,17
Gauss code	{1, -9, 2, -10}, {7, -1, 3, -4, 9, -8}, {5, -2, 6, -7, 8, -6, 10, -5, 4, -3}

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} , $w$, ...)	$-{\frac {(t(1)-1)(t(3)t(2)-t(2)-2t(3)+1)(t(3)t(2)-2t(2)-t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+4q^{4}+4q^{-4}-8q^{3}-8q^{-3}+13q^{2}+13q^{-2}-15q-15q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}-z^{2}a^{-4}+2a^{2}z^{4}+2z^{4}a^{-2}+2a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}-z^{6}-2z^{4}-3z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+6z^{8}a^{-2}+12z^{8}+7a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+7z^{7}a^{-3}+4a^{4}z^{6}-3a^{2}z^{6}-3z^{6}a^{-2}+4z^{6}a^{-4}-14z^{6}+a^{5}z^{5}-11a^{3}z^{5}-28az^{5}-28z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-7a^{2}z^{4}-7z^{4}a^{-2}-6z^{4}a^{-4}-2z^{4}-a^{5}z^{3}+5a^{3}z^{3}+12az^{3}+12z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+3a^{4}z^{2}+5a^{2}z^{2}+5z^{2}a^{-2}+3z^{2}a^{-4}+4z^{2}+2az+2za^{-1}-2a^{2}-2a^{-2}-3-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, 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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   3   3 7                 5 1   -4 5               8 3     5 3             8 6       -2 1           10 7         3 -1         7 10           3 -3       6 8             -2 -5     3 8               5 -7   1 5                 -4 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

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