L11a532

L11a531

L11a533

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a532 at Knotilus!

Link Presentations

[edit Notes on L11a532's Link Presentations]

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

A Braid Representative
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} , 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} , 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} , ...)	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 -\frac{u v w x^3-u v w x^2+u v w x-u v w-u v x^3+2 u v x^2-2 u v x+2 u v-2 u w x^3+2 u w x^2-2 u w x+u w+2 u x^3-3 u x^2+3 u x-2 u-2 v w x^3+3 v w x^2-3 v w x+2 v w+v x^3-2 v x^2+2 v x-2 v+2 w x^3-2 w x^2+2 w x-w-x^3+x^2-x+1}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}}} (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-14q^{5/2}+17q^{3/2}-19{\sqrt {q}}+{\frac {16}{\sqrt {q}}}-{\frac {16}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+12az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-11a^{3}z+22az-15za^{-1}+3za^{-3}+3a^{5}z^{-1}-14a^{3}z^{-1}+22az^{-1}-14a^{-1}z^{-1}+3a^{-3}z^{-1}+2a^{5}z^{-3}-7a^{3}z^{-3}+9az^{-3}-5a^{-1}z^{-3}+a^{-3}z^{-3}$ (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 -a^2 z^{10}-z^{10}-a^3 z^9-6 a z^9-5 z^9 a^{-1} -a^4 z^8-a^2 z^8-11 z^8 a^{-2} -11 z^8-a^5 z^7-3 a^3 z^7+6 a z^7-6 z^7 a^{-1} -14 z^7 a^{-3} +a^4 z^6-2 a^2 z^6+9 z^6 a^{-2} -10 z^6 a^{-4} +16 z^6+6 a^5 z^5+23 a^3 z^5+21 a z^5+28 z^5 a^{-1} +20 z^5 a^{-3} -4 z^5 a^{-5} +10 a^4 z^4+36 a^2 z^4+19 z^4 a^{-2} +10 z^4 a^{-4} -z^4 a^{-6} +34 z^4-14 a^5 z^3-42 a^3 z^3-42 a z^3-22 z^3 a^{-1} -8 z^3 a^{-3} -26 a^4 z^2-73 a^2 z^2-34 z^2 a^{-2} -6 z^2 a^{-4} -75 z^2+16 a^5 z+39 a^3 z+37 a z+16 z a^{-1} +2 z a^{-3} +23 a^4+60 a^2+24 a^{-2} +4 a^{-4} +58-9 a^5 z^{-1} -23 a^3 z^{-1} -24 a z^{-1} -12 a^{-1} z^{-1} -2 a^{-3} z^{-1} -7 a^4 z^{-2} -19 a^2 z^{-2} -7 a^{-2} z^{-2} - a^{-4} z^{-2} -18 z^{-2} +2 a^5 z^{-3} +7 a^3 z^{-3} +9 a z^{-3} +5 a^{-1} z^{-3} + a^{-3} z^{-3} } (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   7 1   -6 6                 7 3     4 4               10 7       -3 2             9 7         2 0           11 14           3 -2         5 5             0 -4       2 11               9 -6     4 5                 -1 -8   1 6                   5 -10                         0 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ 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}} $r=-4$ ${\mathbb {Z} }^{6}$ 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 {\mathbb Z}^{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 r=-3} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{5}$ 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=-1} ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{9}$ $r=1$ 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}^{7}\oplus{\mathbb Z}_2^{10}} ${\mathbb {Z} }^{10}$ 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} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ 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=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^{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 {\mathbb Z}^{7}} $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ 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}} 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=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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