L11a175

L11a174

L11a176

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a175 at Knotilus!

Link Presentations

[edit Notes on L11a175's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X20,12,21,11 X14,6,15,5 X4,21,5,22 X16,7,17,8 X22,15,7,16 X18,14,19,13 X6,18,1,17 X12,20,13,19
Gauss code	{1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -11, 9, -5, 8, -7, 10, -9, 11, -4, 6, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-4t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-2t(2)^{3}+4t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+4t(2)^{2}-2t(1)^{2}t(2)+5t(1)t(2)-4t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}+q^{7/2}-{\frac {12}{q^{7/2}}}-3q^{5/2}+{\frac {13}{q^{5/2}}}+5q^{3/2}-{\frac {13}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {6}{q^{11/2}}}-9{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}-az^{7}+a^{5}z^{5}-4a^{3}z^{5}-4az^{5}+z^{5}a^{-1}+3a^{5}z^{3}-5a^{3}z^{3}-4az^{3}+3z^{3}a^{-1}+2a^{5}z-4a^{3}z+za^{-1}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-9az^{9}-3z^{9}a^{-1}-10a^{4}z^{8}-4a^{2}z^{8}-z^{8}a^{-2}+5z^{8}-11a^{5}z^{7}+11a^{3}z^{7}+38az^{7}+16z^{7}a^{-1}-9a^{6}z^{6}+25a^{4}z^{6}+36a^{2}z^{6}+5z^{6}a^{-2}+7z^{6}-6a^{7}z^{5}+23a^{5}z^{5}+12a^{3}z^{5}-45az^{5}-28z^{5}a^{-1}-3a^{8}z^{4}+12a^{6}z^{4}-14a^{4}z^{4}-43a^{2}z^{4}-8z^{4}a^{-2}-22z^{4}-a^{9}z^{3}+4a^{7}z^{3}-12a^{5}z^{3}-22a^{3}z^{3}+12az^{3}+17z^{3}a^{-1}-3a^{6}z^{2}+4a^{4}z^{2}+13a^{2}z^{2}+4z^{2}a^{-2}+10z^{2}-a^{7}z+4a^{5}z+10a^{3}z+4az-za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (db)

Khovanov Homology

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

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 8                       1 -1 6                     2   2 4                   3 1   -2 2                 6 2     4 0               5 3       -2 -2             8 6         2 -4           6 6           0 -6         6 7             -1 -8       4 7               3 -10     2 5                 -3 -12   1 4                   3 -14   2                     -2 -16 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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