L10a161

L10a160

L10a162

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a161 at Knotilus!

Link Presentations

[edit Notes on L10a161's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{3}-t(1)^{2}t(2)t(3)^{3}-t(1)^{2}t(3)^{2}+t(1)t(2)^{2}t(3)^{2}-t(2)^{2}t(3)^{2}+t(1)^{2}t(2)t(3)^{2}-t(1)t(2)t(3)^{2}+t(1)^{2}t(3)+t(2)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+t(1)+t(2)-1}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{11}-2q^{10}+3q^{9}-4q^{8}+5q^{7}-4q^{6}+5q^{5}-3q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$z^{6}a^{-8}+5z^{4}a^{-8}+7z^{2}a^{-8}+a^{-8}z^{-2}+3a^{-8}-z^{8}a^{-6}-7z^{6}a^{-6}-17z^{4}a^{-6}-18z^{2}a^{-6}-2a^{-6}z^{-2}-9a^{-6}+z^{6}a^{-4}+6z^{4}a^{-4}+11z^{2}a^{-4}+a^{-4}z^{-2}+6a^{-4}$ (db)
Kauffman polynomial	$z^{9}a^{-5}+z^{9}a^{-7}+z^{8}a^{-4}+4z^{8}a^{-6}+3z^{8}a^{-8}-5z^{7}a^{-5}-z^{7}a^{-7}+4z^{7}a^{-9}-7z^{6}a^{-4}-23z^{6}a^{-6}-11z^{6}a^{-8}+5z^{6}a^{-10}+5z^{5}a^{-5}-11z^{5}a^{-7}-12z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-4}+43z^{4}a^{-6}+9z^{4}a^{-8}-14z^{4}a^{-10}+3z^{4}a^{-12}+6z^{3}a^{-5}+20z^{3}a^{-7}+6z^{3}a^{-9}-6z^{3}a^{-11}+2z^{3}a^{-13}-17z^{2}a^{-4}-32z^{2}a^{-6}-3z^{2}a^{-8}+9z^{2}a^{-10}-2z^{2}a^{-12}+z^{2}a^{-14}-9za^{-5}-9za^{-7}+7a^{-4}+11a^{-6}+3a^{-8}-2a^{-10}+2a^{-5}z^{-1}+2a^{-7}z^{-1}-a^{-4}z^{-2}-2a^{-6}z^{-2}-a^{-8}z^{-2}$ (db)

Khovanov Homology

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

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

Integral Khovanov Homology (db, data source)

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

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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