L11a299

L11a298

L11a300

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a299 at Knotilus!

Link Presentations

[edit Notes on L11a299's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, 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 v} , $w$, ...)	$-{\frac {(2t(2)t(1)-t(1)-t(2)+2)\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+t(2)t(1)-t(1)-t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+6q^{19/2}-10q^{17/2}+12q^{15/2}-13q^{13/2}+13q^{11/2}-11q^{9/2}+7q^{7/2}-5q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+2za^{-9}-z^{7}a^{-7}-4z^{5}a^{-7}-5z^{3}a^{-7}-3za^{-7}-z^{7}a^{-5}-4z^{5}a^{-5}-4z^{3}a^{-5}-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	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 z^4 a^{-14} -z^2 a^{-14} +3 z^5 a^{-13} -3 z^3 a^{-13} +z a^{-13} +5 z^6 a^{-12} -5 z^4 a^{-12} +2 z^2 a^{-12} +6 z^7 a^{-11} -7 z^5 a^{-11} +3 z^3 a^{-11} +z a^{-11} +5 z^8 a^{-10} -5 z^6 a^{-10} +z^4 a^{-10} +3 z^9 a^{-9} -z^7 a^{-9} -3 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^{10} a^{-8} +4 z^8 a^{-8} -12 z^6 a^{-8} +8 z^4 a^{-8} -5 z^2 a^{-8} +5 z^9 a^{-7} -13 z^7 a^{-7} +10 z^5 a^{-7} -6 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +z^8 a^{-6} -10 z^6 a^{-6} +9 z^4 a^{-6} -2 z^2 a^{-6} +2 z^9 a^{-5} -5 z^7 a^{-5} -2 z^5 a^{-5} +9 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +2 z^8 a^{-4} -8 z^6 a^{-4} +8 z^4 a^{-4} - a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +8 z^3 a^{-3} -5 z a^{-3} + a^{-3} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     2   2 20                   4 1   -3 18                 6 2     4 16               6 4       -2 14             7 6         1 12           7 7           0 10         4 6             -2 8       3 7               4 6     2 4                 -2 4   1 4                   3 2   1                     -1 0 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=4} $i=6$ $r=-2$ ${\mathbb {Z} }$ 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=-1} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ 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 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}^{4}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ 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 r=2} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ 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}^{6}\oplus{\mathbb Z}_2^{7}} ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ 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}} $r=5$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ 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 r=6} ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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