L11a17

L11a16

L11a18

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a17 at Knotilus!

Link Presentations

[edit Notes on L11a17's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X14,6,15,5 X8493 X16,10,17,9 X10,16,11,15 X20,12,21,11 X22,14,5,13 X12,22,13,21 X2,18,3,17
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 6, -7, 8, -10, 9, -4, 7, -6, 11, -2, 3, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in 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 u} , $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-7q^{15/2}+12q^{13/2}-15q^{11/2}+16q^{9/2}-17q^{7/2}+13q^{5/2}-10q^{3/2}+5{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+4z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-3z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-7}-za^{-1}+6za^{-3}-6za^{-5}+za^{-7}+3a^{-3}z^{-1}-5a^{-5}z^{-1}+2a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-4z^{9}a^{-3}-9z^{9}a^{-5}-5z^{9}a^{-7}-4z^{8}a^{-2}-2z^{8}a^{-6}-6z^{8}a^{-8}-3z^{7}a^{-1}+10z^{7}a^{-3}+28z^{7}a^{-5}+9z^{7}a^{-7}-6z^{7}a^{-9}+11z^{6}a^{-2}+10z^{6}a^{-4}+9z^{6}a^{-6}+7z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+10z^{5}a^{-1}-5z^{5}a^{-3}-37z^{5}a^{-5}-12z^{5}a^{-7}+9z^{5}a^{-9}-z^{5}a^{-11}-6z^{4}a^{-2}-13z^{4}a^{-4}-9z^{4}a^{-6}+2z^{4}a^{-8}+7z^{4}a^{-10}+3z^{4}-8z^{3}a^{-1}-6z^{3}a^{-3}+13z^{3}a^{-5}+9z^{3}a^{-7}-z^{3}a^{-9}+z^{3}a^{-11}-2z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}a^{-8}-2z^{2}a^{-10}-z^{2}+2za^{-1}+8za^{-3}+7za^{-5}+za^{-7}+5a^{-4}+5a^{-6}-a^{-10}-3a^{-3}z^{-1}-5a^{-5}z^{-1}-2a^{-7}z^{-1}$ (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                   4 1   3 14                 8 3     -5 12               7 4       3 10             9 8         -1 8           8 7           1 6         5 9             4 4       5 8               -3 2     2 7                 5 0   1 3                   -2 -2   2                     2 -4 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-3$ ${\mathbb {Z} }$ 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 r=-2} ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ 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^{2}} ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{5}$ 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 r=1} 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}^{8}\oplus{\mathbb Z}_2^{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}^{5}} $r=2$ 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}^{9}\oplus{\mathbb Z}_2^{8}} 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}^{8}} $r=3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ 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}^{8}} $r=6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=8$ 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}_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.

L11a16 L11a18