L11a493

L11a492

L11a494

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Link Presentations

[edit Notes on L11a493's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X10,4,11,3 X18,12,5,11 X20,10,21,9 X22,16,19,15 X14,22,15,21 X2,14,3,13 X8,20,9,19
Gauss code	{1, -10, 5, -3}, {11, -7, 9, -8}, {4, -1, 2, -11, 7, -5, 6, -4, 10, -9, 8, -2, 3, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{(t(1)-1) (t(3)-1) \left(t(3)^2-t(3)+1\right) \left(t(2) t(3)^2-2 t(2) t(3)+2 t(3)-1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}}$ (db)
Jones polynomial	$q^{10}-4 q^9+10 q^8-16 q^7+20 q^6-23 q^5+24 q^4-18 q^3+15 q^2-8 q+4- q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^8 a^{-4} -z^6 a^{-2} +5 z^6 a^{-4} -2 z^6 a^{-6} -3 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} -2 z^2 a^{-2} +9 z^2 a^{-4} -9 z^2 a^{-6} +2 z^2 a^{-8} + a^{-2} +2 a^{-4} -5 a^{-6} +2 a^{-8} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db)
Kauffman polynomial	$2 z^{10} a^{-4} +2 z^{10} a^{-6} +5 z^9 a^{-3} +14 z^9 a^{-5} +9 z^9 a^{-7} +4 z^8 a^{-2} +12 z^8 a^{-4} +24 z^8 a^{-6} +16 z^8 a^{-8} +z^7 a^{-1} -11 z^7 a^{-3} -26 z^7 a^{-5} +2 z^7 a^{-7} +16 z^7 a^{-9} -14 z^6 a^{-2} -56 z^6 a^{-4} -76 z^6 a^{-6} -24 z^6 a^{-8} +10 z^6 a^{-10} -3 z^5 a^{-1} -2 z^5 a^{-3} -11 z^5 a^{-5} -39 z^5 a^{-7} -23 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-2} +66 z^4 a^{-4} +66 z^4 a^{-6} +8 z^4 a^{-8} -8 z^4 a^{-10} +z^4 a^{-12} +3 z^3 a^{-1} +14 z^3 a^{-3} +32 z^3 a^{-5} +34 z^3 a^{-7} +13 z^3 a^{-9} -8 z^2 a^{-2} -29 z^2 a^{-4} -25 z^2 a^{-6} +4 z^2 a^{-10} -z a^{-1} -3 z a^{-3} -11 z a^{-5} -13 z a^{-7} -4 z a^{-9} +4 a^{-4} +5 a^{-6} + a^{-8} - a^{-10} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db)

Khovanov Homology

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

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 21                       1 1 19                     3   -3 17                   7 1   6 15                 9 3     -6 13               11 7       4 11             14 11         -3 9           10 9           1 7         8 14             6 5       7 10               -3 3     3 10                 7 1   1 5                   -4 -1   3                     3 -3 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14}$ ${\mathbb Z}^{14}$ $r=4$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{11}$ $r=5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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