L11a172

L11a171

L11a173

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a172 at Knotilus!

Link Presentations

[edit Notes on L11a172's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X22,16,7,15 X6,21,1,22 X18,11,19,12 X14,6,15,5 X20,13,21,14 X12,19,13,20 X4,18,5,17
Gauss code	{1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -7, 10, -9, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-2t(1)^{2}t(2)^{5}+3t(1)t(2)^{5}-t(2)^{5}+3t(1)^{2}t(2)^{4}-5t(1)t(2)^{4}+2t(2)^{4}-3t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-3t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {15}{q^{7/2}}}-q^{5/2}+{\frac {14}{q^{5/2}}}+3q^{3/2}-{\frac {13}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {10}{q^{11/2}}}-6{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-5a^{5}z^{5}-8a^{5}z^{3}-4a^{5}z+a^{3}z^{9}+7a^{3}z^{7}+18a^{3}z^{5}+20a^{3}z^{3}+8a^{3}z+a^{3}z^{-1}-az^{7}-5az^{5}-8az^{3}-5az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}-a^{10}z^{2}+3a^{9}z^{5}-3a^{9}z^{3}+5a^{8}z^{6}-5a^{8}z^{4}+a^{8}z^{2}+7a^{7}z^{7}-12a^{7}z^{5}+10a^{7}z^{3}-2a^{7}z+7a^{6}z^{8}-14a^{6}z^{6}+12a^{6}z^{4}-2a^{6}z^{2}+5a^{5}z^{9}-9a^{5}z^{7}+5a^{5}z^{5}-2a^{5}z^{3}+2a^{5}z+2a^{4}z^{10}+2a^{4}z^{8}-16a^{4}z^{6}+16a^{4}z^{4}-6a^{4}z^{2}+9a^{3}z^{9}-33a^{3}z^{7}+46a^{3}z^{5}-36a^{3}z^{3}+11a^{3}z-a^{3}z^{-1}+2a^{2}z^{10}-2a^{2}z^{8}-9a^{2}z^{6}+11a^{2}z^{4}-5a^{2}z^{2}+a^{2}+4az^{9}-16az^{7}+z^{7}a^{-1}+22az^{5}-4z^{5}a^{-1}-17az^{3}+4z^{3}a^{-1}+7az-az^{-1}+3z^{8}-12z^{6}+13z^{4}-3z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   4 1   3 0                 5 2     -3 -2               8 4       4 -4             7 6         -1 -6           8 7           1 -8         6 8             2 -10       4 7               -3 -12     2 6                 4 -14   1 4                   -3 -16   2                     2 -18 1                       -1

Integral Khovanov Homology (db, data source)

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

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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