L10a172

L10a171

L10a173

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a172 at Knotilus!

Link Presentations

[edit Notes on L10a172's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(2)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)-t(4)^{2}t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)t(4)^{2}-t(1)+t(1)t(2)-t(2)+2t(1)t(4)-t(1)t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$8q^{9/2}-11q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-11q^{3/2}+{\frac {2}{q^{3/2}}}-q^{15/2}+3q^{13/2}-6q^{11/2}+6{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-a^{-5}z^{-3}-3za^{-5}-2a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+3a^{-3}z^{-3}+11za^{-3}+7a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-11za^{-1}+3az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}+6z^{5}a^{-7}-5z^{3}a^{-7}+2za^{-7}+8z^{6}a^{-6}-10z^{4}a^{-6}+2z^{2}a^{-6}+9z^{7}a^{-5}-20z^{5}a^{-5}+17z^{3}a^{-5}-a^{-5}z^{-3}-12za^{-5}+5a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-17z^{4}a^{-4}+20z^{2}a^{-4}+3a^{-4}z^{-2}-10a^{-4}+z^{9}a^{-3}+12z^{7}a^{-3}-51z^{5}a^{-3}+61z^{3}a^{-3}-3a^{-3}z^{-3}-35za^{-3}+12a^{-3}z^{-1}+7z^{8}a^{-2}-19z^{6}a^{-2}+26z^{2}a^{-2}+6a^{-2}z^{-2}-19a^{-2}+z^{9}a^{-1}+az^{7}+4z^{7}a^{-1}-5az^{5}-30z^{5}a^{-1}+10az^{3}+48z^{3}a^{-1}-az^{-3}-3a^{-1}z^{-3}-10az-31za^{-1}+5az^{-1}+12a^{-1}z^{-1}+2z^{8}-7z^{6}+4z^{4}+8z^{2}+3z^{-2}-10$ (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               5 3     -2 8             6 3       3 6           5 7         2 4         6 4           2 2       4 9             5 0     2 2               0 -2   1 5                 4 -4   1                   -1 -6 1                     1

Integral Khovanov Homology (db, data source)

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

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.

L10a171 L10a173