L11a379

L11a378

L11a380

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a379 at Knotilus!

Link Presentations

[edit Notes on L11a379's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}v^{2}-u^{4}v-u^{3}v^{4}+4u^{3}v^{3}-6u^{3}v^{2}+4u^{3}v-u^{3}+2u^{2}v^{4}-7u^{2}v^{3}+9u^{2}v^{2}-7u^{2}v+2u^{2}-uv^{4}+4uv^{3}-6uv^{2}+4uv-u-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+3q^{3/2}-7{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {21}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {14}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+2a^{5}z^{5}+4a^{5}z^{3}+a^{5}z-a^{3}z^{7}-3a^{3}z^{5}-3a^{3}z^{3}-a^{3}z+a^{3}z^{-1}+2az^{5}+5az^{3}-z^{3}a^{-1}+2az-az^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}-4z^{5}a^{9}+z^{3}a^{9}-9z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-13z^{7}a^{7}+18z^{5}a^{7}-10z^{3}a^{7}+2za^{7}-12z^{8}a^{6}+15z^{6}a^{6}-2z^{4}a^{6}-7z^{9}a^{5}+21z^{5}a^{5}-11z^{3}a^{5}-2z^{10}a^{4}-13z^{8}a^{4}+39z^{6}a^{4}-25z^{4}a^{4}+6z^{2}a^{4}-11z^{9}a^{3}+25z^{7}a^{3}-11z^{5}a^{3}+3z^{3}a^{3}-5za^{3}+a^{3}z^{-1}-2z^{10}a^{2}-4z^{8}a^{2}+26z^{6}a^{2}-27z^{4}a^{2}+8z^{2}a^{2}-a^{2}-4z^{9}a+11z^{7}a-6z^{5}a-2z^{3}a-za+az^{-1}-3z^{8}+11z^{6}-13z^{4}+5z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-5z^{3}a^{-1}+2za^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                       1 1 4                     2   -2 2                   5 1   4 0                 7 2     -5 -2               10 5       5 -4             10 8         -2 -6           11 9           2 -8         8 11             3 -10       6 10               -4 -12     3 8                 5 -14   1 6                   -5 -16   3                     3 -18 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-3$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a378 L11a380