L11a180

L11a179

L11a181

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a180 at Knotilus!

Link Presentations

[edit Notes on L11a180's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-6t(2)^{3}+9t(1)^{2}t(2)^{2}-19t(1)t(2)^{2}+9t(2)^{2}-6t(1)^{2}t(2)+11t(1)t(2)-5t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-4q^{13/2}+10q^{11/2}-18q^{9/2}+24q^{7/2}-29q^{5/2}+29q^{3/2}-26{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+3z^{5}a^{-3}+az^{3}-5z^{3}a^{-1}+5z^{3}a^{-3}-3z^{3}a^{-5}+az-za^{-1}+2za^{-3}-2za^{-5}+za^{-7}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-2z^{4}a^{-8}+z^{2}a^{-8}+4z^{7}a^{-7}-8z^{5}a^{-7}+6z^{3}a^{-7}-2za^{-7}+8z^{8}a^{-6}-16z^{6}a^{-6}+12z^{4}a^{-6}-4z^{2}a^{-6}+8z^{9}a^{-5}-7z^{7}a^{-5}-10z^{5}a^{-5}+12z^{3}a^{-5}-3za^{-5}+3z^{10}a^{-4}+19z^{8}a^{-4}-54z^{6}a^{-4}+41z^{4}a^{-4}-10z^{2}a^{-4}+19z^{9}a^{-3}-23z^{7}a^{-3}+a^{3}z^{5}-12z^{5}a^{-3}+15z^{3}a^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+27z^{8}a^{-2}+5a^{2}z^{6}-65z^{6}a^{-2}-3a^{2}z^{4}+41z^{4}a^{-2}-8z^{2}a^{-2}+a^{-2}+11z^{9}a^{-1}+12az^{7}-15az^{5}-26z^{5}a^{-1}+6az^{3}+15z^{3}a^{-1}-az-a^{-1}z^{-1}+16z^{8}-23z^{6}+11z^{4}-3z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   7 1   -6 10                 11 3     8 8               13 7       -6 6             16 11         5 4           14 14           0 2         12 15             -3 0       8 15               7 -2     4 11                 -7 -4   1 8                   7 -6   4                     -4 -8 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{16}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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