L11a46

L11a45

L11a47

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a46 at Knotilus!

Link Presentations

[edit Notes on L11a46's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X20,17,21,18 X18,10,19,9 X8,20,9,19 X4,21,1,22 X14,12,15,11 X10,4,11,3 X12,5,13,6 X22,13,5,14 X2,16,3,15
Gauss code	{1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, 5, -3, 6, -10}

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} , $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {10}{q^{9/2}}}-5q^{7/2}+{\frac {16}{q^{7/2}}}+10q^{5/2}-{\frac {22}{q^{5/2}}}-17q^{3/2}+{\frac {26}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+22{\sqrt {q}}-{\frac {26}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z-2a^{3}z^{-1}-a^{-3}z^{-1}+az^{7}+3az^{5}-2z^{5}a^{-1}+5az^{3}-3z^{3}a^{-1}+3az+az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+9a^{5}z^{7}-14a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+11a^{4}z^{8}-15a^{4}z^{6}+z^{6}a^{-4}+9a^{4}z^{4}-z^{4}a^{-4}-3a^{4}z^{2}+a^{4}+7a^{3}z^{9}+7a^{3}z^{7}+5z^{7}a^{-3}-35a^{3}z^{5}-10z^{5}a^{-3}+35a^{3}z^{3}+5z^{3}a^{-3}-15a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{10}+22a^{2}z^{8}+9z^{8}a^{-2}-51a^{2}z^{6}-19z^{6}a^{-2}+36a^{2}z^{4}+11z^{4}a^{-2}-12a^{2}z^{2}-2z^{2}a^{-2}+3a^{2}+a^{-2}+14az^{9}+7z^{9}a^{-1}-9az^{7}-2z^{7}a^{-1}-32az^{5}-22z^{5}a^{-1}+36az^{3}+18z^{3}a^{-1}-12az-za^{-1}+az^{-1}-a^{-1}z^{-1}+2z^{10}+20z^{8}-52z^{6}+35z^{4}-10z^{2}+2$ (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 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} ).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 11 4     7 2               11 6       -5 0             15 11         4 -2           13 13           0 -4         9 13             -4 -6       7 13               6 -8     3 9                 -6 -10   1 7                   6 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\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 i=-2} $i=0$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ 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}} $r=-3$ 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^{7}} ${\mathbb {Z} }^{7}$ 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} ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ 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 r=0} ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{15}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ 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}^{4}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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