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L10a167 L10a169

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-13/2 + 2q-11/2 - 6q-9/2 + 9q-7/2 - 14q-5/2 + 12q-3/2 - 14q-1/2 + 10q1/2 - 8q3/2 + 3q5/2 - q7/2

A2 (sl(3)) Invariant:

q-22 + 2q-20 + 2q-16 + 5q-14 + q-12 + 8q-10 + 10q-8 + 10q-6 + 14q-4 + 8q-2 + 11 + 4q2 + 2q4 + 4q6 - 2q8 + q12

HOMFLY-PT Polynomial:

- a-3z - a-1z-3 - 3a-1z-1 - a-1z + 2a-1z3 + 3az-3 + 7az-1 + 5az + az3 - az5 - 3a3z-3 - 4a3z-1 + 3a3z3 + a5z-3 - a5z-1 - 3a5z + a7z-1

Kauffman Polynomial:

- a-3z + 2a-3z3 - a-3z5 - a-2z2 + 4a-2z4 - 3a-2z6 + a-1z-3 - 6a-1z-1 + 14a-1z - 15a-1z3 + 13a-1z5 - 6a-1z7 - 3z-2 + 13 - 22z2 + 20z4 - 2z6 - 4z8 + 3az-3 - 14az-1 + 33az - 45az3 + 37az5 - 13az7 - az9 - 6a2z-2 + 24a2 - 36a2z2 + 21a2z4 + 3a2z6 - 7a2z8 + 3a3z-3 - 12a3z-1 + 24a3z - 32a3z3 + 27a3z5 - 10a3z7 - a3z9 - 3a4z-2 + 11a4 - 15a4z2 + 8a4z4 - 3a4z8 + a5z-3 - 3a5z-1 + 3a5z - a5z3 + 3a5z5 - 3a5z7 - a6 + 3a6z4 - 2a6z6 + a7z-1 - 3a7z + 3a7z3 - a7z5

Khovanov Homology:

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                     1 j = 6                   3 1 j = 4                 5     j = 2               5 3     j = 0             9 5       j = -2           7 9         j = -4         7 5           j = -6       2 7             j = -8     4 7               j = -10   1 5                 j = -12   1                   j = -14 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, 168]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[10, Alternating, 168]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[14, 7, 15, 8], X[8, 13, 5, 14], > X[18, 12, 19, 11], X[20, 16, 17, 15], X[16, 20, 9, 19], X[12, 18, 13, 17], > X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10}, {9, -1, 3, -4}, {8, -5, 7, -6}, > {10, -2, 5, -8, 4, -3, 6, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(13/2) 2 6 9 14 12 14 3/2 -q + ----- - ---- + ---- - ---- + ---- - ------- + 10 Sqrt[q] - 8 q + 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q 5/2 7/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 2 5 -12 8 10 10 14 8 2 4 11 + q + --- + --- + --- + q + --- + -- + -- + -- + -- + 4 q + 2 q + 20 16 14 10 8 6 4 2 q q q q q q q q 6 8 12 > 4 q - 2 q + q
In[8]:=	HOMFLYPT[Link[10, Alternating, 168]][a, z]
Out[8]=	3 5 3 5 7 1 3 a 3 a a 3 7 a 4 a a a z z -(----) + --- - ---- + -- - --- + --- - ---- - -- + -- - -- - - + 5 a z - 3 3 3 3 a z z z z z 3 a a z z z z a 3 5 2 z 3 3 3 5 > 3 a z + ---- + a z + 3 a z - a z a
In[9]:=	Kauffman[Link[10, Alternating, 168]][a, z]
Out[9]=	3 5 2 4 2 4 6 1 3 a 3 a a 3 6 a 3 a 6 13 + 24 a + 11 a - a + ---- + --- + ---- + -- - -- - ---- - ---- - --- - 3 3 3 3 2 2 2 a z a z z z z z z z 3 5 7 14 a 12 a 3 a a z 14 z 3 5 7 > ---- - ----- - ---- + -- - -- + ---- + 33 a z + 24 a z + 3 a z - 3 a z - z z z z 3 a a 2 3 3 2 z 2 2 4 2 2 z 15 z 3 3 3 > 22 z - -- - 36 a z - 15 a z + ---- - ----- - 45 a z - 32 a z - 2 3 a a a 4 5 5 3 7 3 4 4 z 2 4 4 4 6 4 z > a z + 3 a z + 20 z + ---- + 21 a z + 8 a z + 3 a z - -- + 2 3 a a 5 6 13 z 5 3 5 5 5 7 5 6 3 z 2 6 > ----- + 37 a z + 27 a z + 3 a z - a z - 2 z - ---- + 3 a z - a 2 a 7 6 6 6 z 7 3 7 5 7 8 2 8 4 8 > 2 a z - ---- - 13 a z - 10 a z - 3 a z - 4 z - 7 a z - 3 a z - a 9 3 9 > a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 1 1 5 4 7 2 7 9 + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 2 14 6 12 5 10 5 10 4 8 4 8 3 6 3 6 2 q q t q t q t q t q t q t q t q t 7 5 7 2 2 2 4 2 6 3 6 4 > ----- + ---- + ---- + 5 t + 5 q t + 3 q t + 5 q t + 3 q t + q t + 4 2 4 2 q t q t q t 8 4 > q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L10a168

L10a167 L10a169