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L11n449 L11n451

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DrawMorseLink

PD Presentation:

X6172 X3,11,4,10 X11,20,12,21 X13,19,14,22 X21,18,22,9 X17,13,18,12 X15,8,16,5 X7,14,8,15 X19,17,20,16 X2536 X9,1,10,4

Gauss Code:

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

Jones Polynomial:

- 4q-9/2 + 8q-7/2 - 14q-5/2 + 14q-3/2 - 18q-1/2 + 14q1/2 - 13q3/2 + 7q5/2 - 3q7/2 + q9/2

A2 (sl(3)) Invariant:

q-18 + q-16 + 6q-14 + 3q-12 + 8q-10 + 11q-8 + 7q-6 + 15q-4 + 8q-2 + 12 + 7q2 + 2q4 + 4q6 - 3q8 - q14

HOMFLY-PT Polynomial:

a-3z-1 + 2a-3z + a-3z3 - a-1z-3 - 6a-1z-1 - 8a-1z - 6a-1z3 - 2a-1z5 + 3az-3 + 11az-1 + 12az + 8az3 + 4az5 + az7 - 3a3z-3 - 8a3z-1 - 7a3z - 3a3z3 - a3z5 + a5z-3 + 2a5z-1 + a5z

Kauffman Polynomial:

a-4 - 3a-4z2 + 3a-4z4 - a-4z6 - 2a-3z-1 + 6a-3z - 9a-3z3 + 8a-3z5 - 3a-3z7 + 6a-2 - 14a-2z2 + 7a-2z4 + 5a-2z6 - 4a-2z8 + a-1z-3 - 11a-1z-1 + 27a-1z - 42a-1z3 + 39a-1z5 - 10a-1z7 - 2a-1z9 - 3z-2 + 18 - 26z2 + 9z4 + 21z6 - 13z8 + 3az-3 - 18az-1 + 44az - 67az3 + 62az5 - 20az7 - 2az9 - 6a2z-2 + 21a2 - 24a2z2 + 9a2z4 + 9a2z6 - 9a2z8 + 3a3z-3 - 14a3z-1 + 34a3z - 44a3z3 + 31a3z5 - 13a3z7 - 3a4z-2 + 9a4 - 9a4z2 + 4a4z4 - 6a4z6 + a5z-3 - 5a5z-1 + 11a5z - 10a5z3

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 10                   1 j = 8                 2   j = 6               5 1   j = 4             8 2     j = 2           6 5       j = 0         12 8         j = -2       8 12           j = -4     6 6             j = -6   2 8               j = -8 2 6                 j = -10 4

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[11, NonAlternating, 450]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 450]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 11, 4, 10], X[11, 20, 12, 21], X[13, 19, 14, 22], > X[21, 18, 22, 9], X[17, 13, 18, 12], X[15, 8, 16, 5], X[7, 14, 8, 15], > X[19, 17, 20, 16], X[2, 5, 3, 6], X[9, 1, 10, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -8, 7}, {-9, 3, -5, 4}, > {-11, 2, -3, 6, -4, 8, -7, 9, -6, 5}]
In[6]:=	Jones[L][q]
Out[6]=	-4 8 14 14 18 3/2 5/2 7/2 ---- + ---- - ---- + ---- - ------- + 14 Sqrt[q] - 13 q + 7 q - 3 q + 9/2 7/2 5/2 3/2 Sqrt[q] q q q q 9/2 > q
In[7]:=	A2Invariant[L][q]
Out[7]=	-18 -16 6 3 8 11 7 15 8 2 4 6 12 + q + q + --- + --- + --- + -- + -- + -- + -- + 7 q + 2 q + 4 q - 14 12 10 8 6 4 2 q q q q q q q 8 14 > 3 q - q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 450]][a, z]
Out[8]=	3 5 3 5 1 3 a 3 a a 1 6 11 a 8 a 2 a 2 z 8 z -(----) + --- - ---- + -- + ---- - --- + ---- - ---- + ---- + --- - --- + 3 3 3 3 3 a z z z z 3 a a z z z z a z a 3 3 5 3 5 z 6 z 3 3 3 2 z 5 > 12 a z - 7 a z + a z + -- - ---- + 8 a z - 3 a z - ---- + 4 a z - 3 a a a 3 5 7 > a z + a z
In[9]:=	Kauffman[Link[11, NonAlternating, 450]][a, z]
Out[9]=	3 5 2 4 -4 6 2 4 1 3 a 3 a a 3 6 a 3 a 18 + a + -- + 21 a + 9 a + ---- + --- + ---- + -- - -- - ---- - ---- - 2 3 3 3 3 2 2 2 a a z z z z z z z 3 5 2 11 18 a 14 a 5 a 6 z 27 z 3 > ---- - --- - ---- - ----- - ---- + --- + ---- + 44 a z + 34 a z + 3 a z z z z 3 a a z a 2 2 3 3 5 2 3 z 14 z 2 2 4 2 9 z 42 z > 11 a z - 26 z - ---- - ----- - 24 a z - 9 a z - ---- - ----- - 4 2 3 a a a a 4 4 3 3 3 5 3 4 3 z 7 z 2 4 4 4 > 67 a z - 44 a z - 10 a z + 9 z + ---- + ---- + 9 a z + 4 a z + 4 2 a a 5 5 6 6 8 z 39 z 5 3 5 6 z 5 z 2 6 4 6 > ---- + ----- + 62 a z + 31 a z + 21 z - -- + ---- + 9 a z - 6 a z - 3 a 4 2 a a a 7 7 8 9 3 z 10 z 7 3 7 8 4 z 2 8 2 z 9 > ---- - ----- - 20 a z - 13 a z - 13 z - ---- - 9 a z - ---- - 2 a z 3 a 2 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	12 4 2 6 2 8 6 6 8 12 + -- + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + 8 t + 2 10 4 8 4 8 3 6 3 6 2 4 2 4 2 q q t q t q t q t q t q t q t q t 2 2 2 4 2 4 3 6 3 6 4 8 4 10 5 > 6 q t + 5 q t + 8 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n450

L11n449 L11n451