L11a488

L11a487

L11a489

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a488 at Knotilus!

Link Presentations

[edit Notes on L11a488's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X16,10,17,9 X22,18,19,17 X20,14,21,13 X14,20,15,19 X18,22,5,21 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(w-1)^{2}\left(2vw^{2}-vw+v+w^{3}-w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones 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 -q^{11}+3 q^{10}-8 q^9+14 q^8-18 q^7+21 q^6-20 q^5+19 q^4-12 q^3+8 q^2-3 q+1} (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-10}-a^{-10}z^{-2}-2a^{-10}+3z^{4}a^{-8}+8z^{2}a^{-8}+4a^{-8}z^{-2}+9a^{-8}-2z^{6}a^{-6}-7z^{4}a^{-6}-12z^{2}a^{-6}-5a^{-6}z^{-2}-13a^{-6}-z^{6}a^{-4}-z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}z^{-2}+5a^{-4}+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$ (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^{10} a^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +9 z^9 a^{-7} +5 z^9 a^{-9} +5 z^8 a^{-4} +16 z^8 a^{-6} +19 z^8 a^{-8} +8 z^8 a^{-10} +3 z^7 a^{-3} -5 z^7 a^{-7} +4 z^7 a^{-9} +6 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -51 z^6 a^{-6} -56 z^6 a^{-8} -14 z^6 a^{-10} +3 z^6 a^{-12} -7 z^5 a^{-3} -18 z^5 a^{-5} -29 z^5 a^{-7} -28 z^5 a^{-9} -9 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +10 z^4 a^{-4} +64 z^4 a^{-6} +71 z^4 a^{-8} +16 z^4 a^{-10} -4 z^4 a^{-12} +4 z^3 a^{-3} +24 z^3 a^{-5} +50 z^3 a^{-7} +39 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} -11 z^2 a^{-4} -48 z^2 a^{-6} -48 z^2 a^{-8} -13 z^2 a^{-10} +z^2 a^{-12} -17 z a^{-5} -35 z a^{-7} -23 z a^{-9} -4 z a^{-11} +z a^{-13} - a^{-2} +8 a^{-4} +22 a^{-6} +19 a^{-8} +5 a^{-10} +5 a^{-5} z^{-1} +9 a^{-7} z^{-1} +5 a^{-9} z^{-1} + a^{-11} z^{-1} -2 a^{-4} z^{-2} -5 a^{-6} z^{-2} -4 a^{-8} z^{-2} - a^{-10} z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums 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 \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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 23                       1 -1 21                     2   2 19                   6 1   -5 17                 8 2     6 15               10 6       -4 13             11 8         3 11           11 12           1 9         8 9             -1 7       4 11               7 5     4 8                 -4 3   1 6                   5 1   2                     -2 -1 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=0} ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{4}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ 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}} $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=5$ 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}^{8}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ $r=6$ 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}^{6}\oplus{\mathbb Z}_2^{8}} ${\mathbb {Z} }^{8}$ $r=7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$ 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}^{6}} $r=8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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