L10a145

L10a144

L10a146

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a145 at Knotilus!

Link Presentations

[edit Notes on L10a145's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
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} , ...)	$-{\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}+t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}+t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)+t(1)-t(1)t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (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^{11}-2 q^{10}+3 q^9-3 q^8+4 q^7-4 q^6+4 q^5-2 q^4+3 q^3-q^2+q} (db)
Signature	6 (db)
HOMFLY-PT 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^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +5 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +7 z^2 a^{-8} +7 a^{-4} -11 a^{-6} +4 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} } (db)
Kauffman polynomial	$z^{2}a^{-14}+2z^{3}a^{-13}+3z^{4}a^{-12}-3z^{2}a^{-12}+a^{-12}+3z^{5}a^{-11}-4z^{3}a^{-11}+3z^{6}a^{-10}-6z^{4}a^{-10}+3z^{7}a^{-9}-9z^{5}a^{-9}+4z^{3}a^{-9}+3z^{8}a^{-8}-14z^{6}a^{-8}+20z^{4}a^{-8}-13z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}-2z^{7}a^{-7}-8z^{5}a^{-7}+19z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+4z^{8}a^{-6}-24z^{6}a^{-6}+46z^{4}a^{-6}-35z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+4z^{5}a^{-5}+9z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}$ (db)

Khovanov Homology

The coefficients of the monomials 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 t^rq^j} are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 23                     1 1 21                   2 1 -1 19                 1     1 17               2 2     0 15             2 1       1 13           3 3         0 11         1 1           0 9       1 3             2 7     2 1               1 5   1 3                 2 3                       0 1 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=5} $i=7$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{3}$ 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 {\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} ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ 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 {\mathbb Z}} $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ 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=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 {\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}}

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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