L10a41

L10a40

L10a42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a41 at Knotilus!

Link Presentations

[edit Notes on L10a41's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, 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} , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (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^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+5 q^{5/2}-\frac{7}{q^{5/2}}-7 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+8 \sqrt{q}-\frac{10}{\sqrt{q}}} (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{5}-a^{5}z^{-1}+2z^{3}a^{3}+4za^{3}+3a^{3}z^{-1}-z^{5}a-2z^{3}a-3za-2az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{5}-3a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}+2a^{4}z^{6}+z^{6}a^{-4}+a^{4}z^{4}-3z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}+3a^{4}+2a^{3}z^{7}+3z^{7}a^{-3}+2a^{3}z^{5}-10z^{5}a^{-3}-7a^{3}z^{3}+8z^{3}a^{-3}+7a^{3}z-2za^{-3}-3a^{3}z^{-1}+2a^{2}z^{8}+3z^{8}a^{-2}-a^{2}z^{6}-8z^{6}a^{-2}+2a^{2}z^{4}+3z^{4}a^{-2}-4a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+az^{9}+z^{9}a^{-1}+2az^{7}+3z^{7}a^{-1}-5az^{5}-15z^{5}a^{-1}+12z^{3}a^{-1}+5az-2za^{-1}-2az^{-1}+5z^{8}-12z^{6}+8z^{4}-2z^{2}$ (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 $r$).

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   2   2 6                 3 1   -2 4               4 2     2 2             4 3       -1 0           6 4         2 -2         4 6           2 -4       3 4             -1 -6     1 4               3 -8   1 3                 -2 -10   1                   1 -12 1                     -1

Integral Khovanov Homology (db, data source)

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

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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