L11a295

L11a294

L11a296

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a295 at Knotilus!

Link Presentations

[edit Notes on L11a295's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X22,20,9,19 X18,7,19,8 X6,15,7,16 X16,5,17,6 X4,17,5,18 X14,22,15,21 X20,14,21,13 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -7, 6, -5, 4, -11}, {10, -1, 11, -2, 9, -8, 5, -6, 7, -4, 3, -9, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u+v)\left(u^{2}v^{3}-2u^{2}v^{2}+2u^{2}v+uv^{4}-4uv^{3}+6uv^{2}-4uv+u+2v^{3}-2v^{2}+v\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {15}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {16}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {9}{q^{11/2}}}-{\frac {6}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+2za^{7}+2a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-4za^{5}-3a^{5}z^{-1}-2z^{5}a^{3}-4z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{5}a+za+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{6}z^{10}-a^{4}z^{10}-3a^{7}z^{9}-7a^{5}z^{9}-4a^{3}z^{9}-2a^{8}z^{8}-5a^{6}z^{8}-11a^{4}z^{8}-8a^{2}z^{8}-a^{9}z^{7}+11a^{7}z^{7}+17a^{5}z^{7}-5a^{3}z^{7}-10az^{7}+7a^{8}z^{6}+25a^{6}z^{6}+33a^{4}z^{6}+7a^{2}z^{6}-8z^{6}+5a^{9}z^{5}-16a^{7}z^{5}-15a^{5}z^{5}+23a^{3}z^{5}+13az^{5}-4z^{5}a^{-1}-6a^{8}z^{4}-30a^{6}z^{4}-30a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+8z^{4}-8a^{9}z^{3}+16a^{7}z^{3}+18a^{5}z^{3}-14a^{3}z^{3}-6az^{3}+2z^{3}a^{-1}+14a^{6}z^{2}+15a^{4}z^{2}-a^{2}z^{2}-2z^{2}+4a^{9}z-10a^{7}z-14a^{5}z+a^{3}z+az-3a^{6}-3a^{4}-a^{2}+2a^{7}z^{-1}+3a^{5}z^{-1}+a^{3}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                       1 -1 4                     3   3 2                   5 1   -4 0                 7 3     4 -2               9 6       -3 -4             8 6         2 -6           8 9           1 -8         5 8             -3 -10       4 8               4 -12     2 5                 -3 -14     4                   4 -16 1 2                     -1 -18 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a294 L11a296