L11a235

L11a234

L11a236

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a235 at Knotilus!

Link Presentations

[edit Notes on L11a235's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,13,21,14 X16,9,17,10 X10,19,11,20 X22,15,7,16 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(2)^{6}-3t(1)t(2)^{5}+3t(2)^{5}-3t(1)^{2}t(2)^{4}+6t(1)t(2)^{4}-4t(2)^{4}+4t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+4t(2)^{3}-4t(1)^{2}t(2)^{2}+6t(1)t(2)^{2}-3t(2)^{2}+3t(1)^{2}t(2)-3t(1)t(2)-t(1)^{2}}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {16}{q^{13/2}}}+{\frac {17}{q^{15/2}}}-{\frac {18}{q^{17/2}}}+{\frac {15}{q^{19/2}}}-{\frac {11}{q^{21/2}}}+{\frac {7}{q^{23/2}}}-{\frac {3}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+4a^{11}z^{3}+10a^{11}z+5a^{11}z^{-1}-3a^{9}z^{5}-9a^{9}z^{3}-8a^{9}z-3a^{9}z^{-1}-3a^{7}z^{5}-8a^{7}z^{3}-4a^{7}z-a^{5}z^{5}-2a^{5}z^{3}$ (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^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3+4 a^{14} z^8-7 a^{14} z^6-2 a^{14} z^4+4 a^{14} z^2+4 a^{13} z^9-7 a^{13} z^7+5 a^{13} z^5-10 a^{13} z^3+7 a^{13} z-2 a^{13} z^{-1} +2 a^{12} z^{10}+2 a^{12} z^8-12 a^{12} z^6+17 a^{12} z^4-17 a^{12} z^2+5 a^{12}+10 a^{11} z^9-31 a^{11} z^7+51 a^{11} z^5-47 a^{11} z^3+22 a^{11} z-5 a^{11} z^{-1} +2 a^{10} z^{10}+5 a^{10} z^8-23 a^{10} z^6+36 a^{10} z^4-21 a^{10} z^2+5 a^{10}+6 a^9 z^9-15 a^9 z^7+23 a^9 z^5-18 a^9 z^3+11 a^9 z-3 a^9 z^{-1} +7 a^8 z^8-16 a^8 z^6+15 a^8 z^4-3 a^8 z^2+6 a^7 z^7-14 a^7 z^5+12 a^7 z^3-4 a^7 z+3 a^6 z^6-5 a^6 z^4+a^5 z^5-2 a^5 z^3} (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 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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   4     4 -10                 7 3     -4 -12               9 4       5 -14             8 7         -1 -16           10 9           1 -18         6 9             3 -20       5 9               -4 -22     2 6                 4 -24   1 5                   -4 -26   2                     2 -28 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}} 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=-6} $i=-4$ $r=-11$ ${\mathbb {Z} }$ $r=-10$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-9$ 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 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{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}^{2}} $r=-8$ 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}^{6}\oplus{\mathbb Z}_2^{5}} ${\mathbb {Z} }^{5}$ $r=-7$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-6$ 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}^{9}\oplus{\mathbb Z}_2^{9}} ${\mathbb {Z} }^{10}$ $r=-5$ 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}^{9}\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} ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ 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}} $r=0$ 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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