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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X15,2,16,3

Gauss Code:

{{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, -8, 7, -10, 8}}

Jones Polynomial:

2q-19/2 - 4q-17/2 + 5q-15/2 - 7q-13/2 + 7q-11/2 - 7q-9/2 + 5q-7/2 - 4q-5/2 + 2q-3/2 - q-1/2

A2 (sl(3)) Invariant:

- q-34 - q-32 - q-30 - q-28 + 2q-26 + q-24 + 3q-22 + q-20 + 2q-16 - q-14 + 2q-12 + q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

- a3z-1 - 4a3z - 4a3z3 - a3z5 + 2a5z-1 + 6a5z + 8a5z3 + 5a5z5 + a5z7 - 3a7z-1 - 8a7z - 8a7z3 - 2a7z5 + 3a9z-1 + 4a9z + a9z3 - a11z-1

Kauffman Polynomial:

- a3z-1 + 5a3z - 8a3z3 + 5a3z5 - a3z7 + 3a4z2 - 11a4z4 + 9a4z6 - 2a4z8 - 2a5z-1 + 11a5z - 23a5z3 + 14a5z5 - a5z9 - 2a6 + 9a6z2 - 20a6z4 + 19a6z6 - 5a6z8 - 3a7z-1 + 15a7z - 27a7z3 + 21a7z5 - 3a7z7 - a7z9 + 3a8z2 - 4a8z4 + 7a8z6 - 3a8z8 - 3a9z-1 + 12a9z - 15a9z3 + 11a9z5 - 4a9z7 + 2a10 - 6a10z2 + 5a10z4 - 3a10z6 - a11z-1 + 3a11z - 3a11z3 - a11z5 + a12 - 3a12z2

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 0                   1 j = -2                 1   j = -4               3 1   j = -6             3 2     j = -8           4 2       j = -10         3 3         j = -12       4 4           j = -14     2 4             j = -16   2 3               j = -18   2                 j = -20 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 7]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 7]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[4, 17, 1, 18], X[9, 14, 10, 15], > X[3, 8, 4, 9], X[5, 11, 6, 10], X[11, 20, 12, 21], X[19, 22, 20, 5], > X[13, 19, 14, 18], X[21, 12, 22, 13], X[15, 2, 16, 3]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, -11, -2, 3, 9, > -8, 7, -10, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 5 7 7 7 5 4 2 1 ----- - ----- + ----- - ----- + ----- - ---- + ---- - ---- + ---- - ------- 19/2 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -32 -30 -28 2 -24 3 -20 2 -14 2 -8 -q - q - q - q + --- + q + --- + q + --- - q + --- + q + 26 22 16 12 q q q q -6 -2 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 7]][a, z]
Out[8]=	3 5 7 9 11 a 2 a 3 a 3 a a 3 5 7 9 -(--) + ---- - ---- + ---- - --- - 4 a z + 6 a z - 8 a z + 4 a z - z z z z z 3 3 5 3 7 3 9 3 3 5 5 5 7 5 5 7 > 4 a z + 8 a z - 8 a z + a z - a z + 5 a z - 2 a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 7]][a, z]
Out[9]=	3 5 7 9 11 6 10 12 a 2 a 3 a 3 a a 3 5 -2 a + 2 a + a - -- - ---- - ---- - ---- - --- + 5 a z + 11 a z + z z z z z 7 9 11 4 2 6 2 8 2 10 2 > 15 a z + 12 a z + 3 a z + 3 a z + 9 a z + 3 a z - 6 a z - 12 2 3 3 5 3 7 3 9 3 11 3 4 4 > 3 a z - 8 a z - 23 a z - 27 a z - 15 a z - 3 a z - 11 a z - 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 20 a z - 4 a z + 5 a z + 5 a z + 14 a z + 21 a z + 11 a z - 11 5 4 6 6 6 8 6 10 6 3 7 7 7 > a z + 9 a z + 19 a z + 7 a z - 3 a z - a z - 3 a z - 9 7 4 8 6 8 8 8 5 9 7 9 > 4 a z - 2 a z - 5 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 2 2 2 3 2 4 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 4 20 7 18 6 16 6 16 5 14 5 14 4 12 4 q q q t q t q t q t q t q t q t 4 3 3 4 2 3 t t 2 > ------ + ------ + ------ + ----- + ---- + ---- + -- + -- + t 12 3 10 3 10 2 8 2 8 6 4 2 q t q t q t q t q t q t q q

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