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L10a170 L10a172

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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X10,13,5,14 X20,16,17,15 X14,7,15,8 X16,20,11,19 X8,17,9,18 X18,9,19,10 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

- q-19/2 + 3q-17/2 - 7q-15/2 + 9q-13/2 - 14q-11/2 + 12q-9/2 - 14q-7/2 + 9q-5/2 - 7q-3/2 + 3q-1/2 - q1/2

A2 (sl(3)) Invariant:

q-30 + q-28 + 6q-24 + 6q-22 + 8q-20 + 14q-18 + 10q-16 + 13q-14 + 8q-12 + 6q-10 + 6q-8 - q-6 + 3q-4 - 1 + q2

HOMFLY-PT Polynomial:

- az - az3 - a3z-3 - 4a3z-1 - 4a3z + a3z5 + 3a5z-3 + 9a5z-1 + 10a5z + 6a5z3 + 2a5z5 - 3a7z-3 - 6a7z-1 - 6a7z - 3a7z3 + a9z-3 + a9z-1 + a9z

Kauffman Polynomial:

- az + 2az3 - az5 - a2z2 + 5a2z4 - 3a2z6 + a3z-3 - 5a3z-1 + 11a3z - 10a3z3 + 10a3z5 - 5a3z7 - 3a4z-2 + 10a4 - 14a4z2 + 9a4z4 + 2a4z6 - 4a4z8 + 3a5z-3 - 12a5z-1 + 33a5z - 54a5z3 + 42a5z5 - 13a5z7 - a5z9 - 6a6z-2 + 19a6 - 26a6z2 + 8a6z4 + 10a6z6 - 8a6z8 + 3a7z-3 - 12a7z-1 + 33a7z - 54a7z3 + 42a7z5 - 13a7z7 - a7z9 - 3a8z-2 + 10a8 - 14a8z2 + 9a8z4 + 2a8z6 - 4a8z8 + a9z-3 - 5a9z-1 + 11a9z - 10a9z3 + 10a9z5 - 5a9z7 - a10z2 + 5a10z4 - 3a10z6 - a11z + 2a11z3 - a11z5

Khovanov Homology:

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

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]=	4
In[3]:=	Show[DrawMorseLink[Link[10, Alternating, 171]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 171]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[20, 16, 17, 15], > X[14, 7, 15, 8], X[16, 20, 11, 19], X[8, 17, 9, 18], X[18, 9, 19, 10], > X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {7, -8, 6, -4}, {9, -1, 5, -7, 8, -3}, > {10, -2, 3, -5, 4, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(19/2) 3 7 9 14 12 14 9 7 -q + ----- - ----- + ----- - ----- + ---- - ---- + ---- - ---- + 17/2 15/2 13/2 11/2 9/2 7/2 5/2 3/2 q q q q q q q q 3 > ------- - Sqrt[q] Sqrt[q]
In[7]:=	A2Invariant[L][q]
Out[7]=	-30 -28 6 6 8 14 10 13 8 6 6 -6 -1 + q + q + --- + --- + --- + --- + --- + --- + --- + --- + -- - q + 24 22 20 18 16 14 12 10 8 q q q q q q q q q 3 2 > -- + q 4 q
In[8]:=	HOMFLYPT[Link[10, Alternating, 171]][a, z]
Out[8]=	3 5 7 9 3 5 7 9 a 3 a 3 a a 4 a 9 a 6 a a 3 5 -(--) + ---- - ---- + -- - ---- + ---- - ---- + -- - a z - 4 a z + 10 a z - 3 3 3 3 z z z z z z z z 7 9 3 5 3 7 3 3 5 5 5 > 6 a z + a z - a z + 6 a z - 3 a z + a z + 2 a z
In[9]:=	Kauffman[Link[10, Alternating, 171]][a, z]
Out[9]=	3 5 7 9 4 6 8 3 4 6 8 a 3 a 3 a a 3 a 6 a 3 a 5 a 10 a + 19 a + 10 a + -- + ---- + ---- + -- - ---- - ---- - ---- - ---- - 3 3 3 3 2 2 2 z z z z z z z z 5 7 9 12 a 12 a 5 a 3 5 7 9 > ----- - ----- - ---- - a z + 11 a z + 33 a z + 33 a z + 11 a z - z z z 11 2 2 4 2 6 2 8 2 10 2 3 > a z - a z - 14 a z - 26 a z - 14 a z - a z + 2 a z - 3 3 5 3 7 3 9 3 11 3 2 4 4 4 > 10 a z - 54 a z - 54 a z - 10 a z + 2 a z + 5 a z + 9 a z + 6 4 8 4 10 4 5 3 5 5 5 7 5 > 8 a z + 9 a z + 5 a z - a z + 10 a z + 42 a z + 42 a z + 9 5 11 5 2 6 4 6 6 6 8 6 10 6 > 10 a z - a z - 3 a z + 2 a z + 10 a z + 2 a z - 3 a z - 3 7 5 7 7 7 9 7 4 8 6 8 8 8 > 5 a z - 13 a z - 13 a z - 5 a z - 4 a z - 8 a z - 4 a z - 5 9 7 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 2 1 5 4 6 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 20 8 18 7 16 7 16 6 14 6 14 5 12 5 q q q t q t q t q t q t q t q t 8 8 6 6 8 8 3 6 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + 2 t + -- + 12 4 10 4 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q t q t q 2 2 > q t

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