L10a164

L10a163

L10a165

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a164 at Knotilus!

Link Presentations

[edit Notes on L10a164's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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 w} , ...)	${\frac {t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}-t(1)^{2}t(3)^{2}+2t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}+2t(1)^{2}t(2)t(3)^{2}-4t(1)t(2)t(3)^{2}+3t(2)t(3)^{2}-t(3)^{2}+2t(1)^{2}t(3)+t(1)^{2}t(2)^{2}t(3)-2t(1)t(2)^{2}t(3)+t(2)^{2}t(3)-2t(1)t(3)-3t(1)^{2}t(2)t(3)+4t(1)t(2)t(3)-2t(2)t(3)-t(1)^{2}+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-2}-3q^{-3}+7q^{-4}-10q^{-5}+13q^{-6}-13q^{-7}+13q^{-8}-9q^{-9}+7q^{-10}-3q^{-11}+q^{-12}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-4a^{10}z^{2}-2a^{10}z^{-2}-6a^{10}+3a^{8}z^{4}+6a^{8}z^{2}+a^{8}z^{-2}+4a^{8}+3a^{6}z^{4}+5a^{6}z^{2}+a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$z^{6}a^{14}-3z^{4}a^{14}+3z^{2}a^{14}-a^{14}+3z^{7}a^{13}-8z^{5}a^{13}+5z^{3}a^{13}+4z^{8}a^{12}-9z^{6}a^{12}+4z^{4}a^{12}-3z^{2}a^{12}-a^{12}z^{-2}+5a^{12}+2z^{9}a^{11}+3z^{7}a^{11}-18z^{5}a^{11}+17z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}+10z^{8}a^{10}-25z^{6}a^{10}+26z^{4}a^{10}-23z^{2}a^{10}-2a^{10}z^{-2}+11a^{10}+2z^{9}a^{9}+7z^{7}a^{9}-22z^{5}a^{9}+19z^{3}a^{9}-9za^{9}+2a^{9}z^{-1}+6z^{8}a^{8}-9z^{6}a^{8}+10z^{4}a^{8}-10z^{2}a^{8}-a^{8}z^{-2}+5a^{8}+7z^{7}a^{7}-9z^{5}a^{7}+5z^{3}a^{7}+6z^{6}a^{6}-8z^{4}a^{6}+6z^{2}a^{6}-a^{6}+3z^{5}a^{5}-2z^{3}a^{5}+z^{4}a^{4}-z^{2}a^{4}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                     1 1 -5                   3 1 -2 -7                 4     4 -9               6 3     -3 -11             7 4       3 -13           6 6         0 -15         7 7           0 -17       4 8             4 -19     3 5               -2 -21   1 5                 4 -23   2                   -2 -25 1                     1

Integral Khovanov Homology (db, data source)

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

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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