L10a43

L10a42

L10a44

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a43 at Knotilus!

Link Presentations

[edit Notes on L10a43's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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} , $v$, 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 w} , ...)	$-{\frac {uv^{5}-4uv^{4}+8uv^{3}-8uv^{2}+4uv+4v^{4}-8v^{3}+8v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (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 -9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-16 q^{5/2}+\frac{3}{q^{5/2}}+17 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+5 q^{11/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}} (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+2z^{5}a^{-3}+3az^{3}-12z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+5az-12za^{-1}+5za^{-3}+3az^{-1}-5a^{-1}z^{-1}+2a^{-3}z^{-1}$ (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^5 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} + a^{-6} +9 z^7 a^{-5} -15 z^5 a^{-5} +5 z^3 a^{-5} +7 z^8 a^{-4} -4 z^6 a^{-4} -9 z^4 a^{-4} +4 z^2 a^{-4} +2 z^9 a^{-3} +15 z^7 a^{-3} +a^3 z^5-34 z^5 a^{-3} -2 a^3 z^3+18 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +12 z^8 a^{-2} +3 a^2 z^6-13 z^6 a^{-2} -4 a^2 z^4-10 z^4 a^{-2} +a^2 z^2+17 z^2 a^{-2} -5 a^{-2} +2 z^9 a^{-1} +6 a z^7+12 z^7 a^{-1} -12 a z^5-31 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -10 a z-16 z a^{-1} +3 a z^{-1} +5 a^{-1} z^{-1} +5 z^8-z^6-11 z^4+14 z^2-5} (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 $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   4   -4 10                 5 1   4 8               8 4     -4 6             8 5       3 4           9 8         -1 2         7 8           -1 0       5 10             5 -2     3 6               -3 -4     5                 5 -6 1 3                   -2 -8 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}} $i=0$ $i=2$ $r=-4$ 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}} 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}} $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ 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}^{5}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ $r=-1$ 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}^{6}\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}} 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} ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{7}$ 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=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^{9}} ${\mathbb {Z} }^{9}$ 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} ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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 r=3} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ 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 r=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}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=5$ 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}\oplus{\mathbb Z}_2^{4}} ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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