L11a173

L11a172

L11a174

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a173 at Knotilus!

Link Presentations

[edit Notes on L11a173's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {\left(t(1)t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-t(2)-t(1)+1\right)\left(t(1)t(2)^{3}-t(2)^{3}-t(1)t(2)^{2}+t(2)^{2}+t(1)t(2)-t(2)+1\right)}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+13{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {14}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+5a^{3}z^{5}+8a^{3}z^{3}+4a^{3}z+a^{3}z^{-1}-az^{9}-7az^{7}+z^{7}a^{-1}-18az^{5}+5z^{5}a^{-1}-20az^{3}+8z^{3}a^{-1}-9az+4za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-9az^{9}-4z^{9}a^{-1}-6a^{4}z^{8}-a^{2}z^{8}-4z^{8}a^{-2}+z^{8}-5a^{5}z^{7}+13a^{3}z^{7}+32az^{7}+11z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+15a^{4}z^{6}+12a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+5z^{6}-a^{7}z^{5}+11a^{5}z^{5}-17a^{3}z^{5}-53az^{5}-15z^{5}a^{-1}+9z^{5}a^{-3}+6a^{6}z^{4}-15a^{4}z^{4}-21a^{2}z^{4}-5z^{4}a^{-2}+3z^{4}a^{-4}-8z^{4}+2a^{7}z^{3}-7a^{5}z^{3}+7a^{3}z^{3}+39az^{3}+17z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+4a^{4}z^{2}+8a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+6z^{2}+2a^{5}z-3a^{3}z-11az-6za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     2   2 6                   4 1   -3 4                 6 2     4 2               7 4       -3 0             9 6         3 -2           7 8           1 -4         7 8             -1 -6       4 8               4 -8     2 6                 -4 -10   1 4                   3 -12   2                     -2 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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