L11a179

L11a178

L11a180

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a179 at Knotilus!

Link Presentations

[edit Notes on L11a179's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(v^{2}-v+1\right)(uv-u-v+2)(2uv-u-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$24q^{9/2}-25q^{7/2}+20q^{5/2}-16q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-9q^{15/2}+16q^{13/2}-21q^{11/2}+9{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+3z^{5}a^{-5}+3z^{3}a^{-5}-za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+2a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-5z^{4}a^{-10}+z^{2}a^{-10}+8z^{7}a^{-9}-11z^{5}a^{-9}+4z^{3}a^{-9}-za^{-9}+11z^{8}a^{-8}-19z^{6}a^{-8}+14z^{4}a^{-8}-4z^{2}a^{-8}-a^{-8}+9z^{9}a^{-7}-11z^{7}a^{-7}+2z^{5}a^{-7}+4z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+3z^{10}a^{-6}+14z^{8}a^{-6}-44z^{6}a^{-6}+41z^{4}a^{-6}-9z^{2}a^{-6}-3a^{-6}+16z^{9}a^{-5}-32z^{7}a^{-5}+17z^{5}a^{-5}+3z^{3}a^{-5}-5za^{-5}+3a^{-5}z^{-1}+3z^{10}a^{-4}+10z^{8}a^{-4}-36z^{6}a^{-4}+30z^{4}a^{-4}-6z^{2}a^{-4}-3a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}-6z^{5}a^{-3}+10z^{3}a^{-3}-6za^{-3}+2a^{-3}z^{-1}+7z^{8}a^{-2}-14z^{6}a^{-2}+6z^{4}a^{-2}-z^{2}a^{-2}+4z^{7}a^{-1}-9z^{5}a^{-1}+6z^{3}a^{-1}-2za^{-1}+z^{6}-2z^{4}+z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over $r$).

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 20                       1 1 18                     3   -3 16                   6 1   5 14                 10 3     -7 12               11 6       5 10             13 10         -3 8           12 11           1 6         9 14             5 4       7 11               -4 2     3 10                 7 0   1 6                   -5 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=2} $i=4$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2} ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{7}} $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{13}} $r=4$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=5$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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