L10a167

L10a166

L10a168

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a167 at Knotilus!

Link Presentations

[edit Notes on L10a167's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X8,18,5,17 X16,8,17,7 X20,14,15,13 X14,16,9,15 X12,20,13,19
Gauss code	{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -10, 8, -9}, {9, -7, 6, -3, 10, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}+t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)+t(1)t(3)+t(2)t(3)-2t(1)t(4)t(3)+t(1)t(2)t(4)t(3)-2t(2)t(4)t(3)+t(4)t(3)-t(3)-2t(1)+t(1)t(2)-t(2)+t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$7q^{9/2}-10q^{7/2}+8q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {1}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+5{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}+az^{3}-9z^{3}a^{-1}+10z^{3}a^{-3}-3z^{3}a^{-5}+4az-14za^{-1}+13za^{-3}-3za^{-5}+4az^{-1}-11a^{-1}z^{-1}+10a^{-3}z^{-1}-3a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-az^{7}-6z^{7}a^{-3}-7z^{7}a^{-5}+13z^{6}a^{-2}+4z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}+6az^{5}+14z^{5}a^{-1}+27z^{5}a^{-3}+13z^{5}a^{-5}-6z^{5}a^{-7}+6z^{4}a^{-2}+11z^{4}a^{-4}+8z^{4}a^{-6}-3z^{4}a^{-8}+6z^{4}-13az^{3}-30z^{3}a^{-1}-30z^{3}a^{-3}-6z^{3}a^{-5}+6z^{3}a^{-7}-z^{3}a^{-9}-33z^{2}a^{-2}-16z^{2}a^{-4}-17z^{2}+13az+28za^{-1}+21za^{-3}+3za^{-5}-3za^{-7}+24a^{-2}+11a^{-4}-a^{-6}+13-6az^{-1}-14a^{-1}z^{-1}-12a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                     1 1 14                   3 1 -2 12                 3     3 10               4 3     -1 8             6 3       3 6           5 7         2 4         4 3           1 2       4 8             4 0     1 1               0 -2     4                 4 -4 1 1                   0 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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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